THE 


AMERICAN  HOUSE-CARPENTER 


A    TREATISE 


THE    ART    OF    BUILDING, 


THE  STRENGTH  OF  MATERIALS. 


BY 

R.    G.    HATFIELD,    ARCHITECT, 

MEir.    AM.   INST.   OF  AECHITECTS. 


SEVENTH      EDITION,      REVISED      AND      ENLARGED 

WITH    ADDITIONAL    ILLUSTRATIONS. 


NEW  YORK: 
JOHN    WILEY    &    SON, 

18  ASTOR  PLACE. 

1873. 


Entered  according  to  Act  of  Congress,  in  the  year  18BT,  by 
R.  G.  HATFIELD, 

te  thft  Cl«rk'»  Office  of  the  District  Court  of  the  United  States,  for  the  Southern  District 
of  New  York. 


PREFACE. 


THIS  book  is  intended  for  carpenters — for  masters, 
journeymen  and  apprentices.  It  has  long  been  the  com- 
plaint of  this  class  that  architectural  books,  intended  for 
their  instruction,  are  of  a  price  so  high  as  to  be  placed 
beyond  their  reach.  This  is  owing,  in  a  great  measure, 
to  the  costliness  of  the  plates  with  which  they  are  illus- 
trated: an  unnecessary  expense,  as  illustrations  upon 
wood,  printed  on  good  paper,  answer  every  useful  pur- 
pose. Wood  engravings,  too,  can  be  distributed  among 
the  letter-press  ;  an  advantage  which  plates  but  partially 
possess,  and  one  of  great  importance  to  the  reader. 

Considerations  of  this  kind  induced  the  author  to 
undertake  the  preparation  of  this  volume.  The  subject 
matter  has  been  gleaned  from  works  of  the  first  autho- 
rity, and  subjected  to  the  most  careful  examination. 
The  explanations  have  all  been  written  out  from  the 
figures  themselves,  and  not  taken  from  any  other  work ; 
and  the  figures  have  all  been  drawn  expressly  for  this 
book.  In  doing  this,  the  utmost  care  has  been  taken  to 
make  everything  as  plain  as  the  nature  of  the  case 
would  admit. 

The  attention  of  the  reader  is  particularly  directed  to 
the  following  new  inventions,  viz ;  an  easy  method  of 
describing  the  curves  of  mouldings  through  three  given 


IV  PREFACE. 

points ;  a  rule  to  determine  the  projection  of  eave  cor 
nices ;  a  new  method  of  proportioning  a  cornice  to  a 
larger  given  one  ;  a  way  to  determine  the  lengths  and 
bevils  of  rafters  for  hip-roofs ;  a  way  to  proportion  the 
rise  to  the  tread  ii  stairs ;  to  determine  the  true  position 
of  butt-joints  in  hand-rails  ;  to  find  the  bevils  for  splayed- 
work ;  a  general  rule  for  scrolls,  &c.  Many  problems  in 
geometry,  also,  have  been  simplified,  and  new  ones  intro- 
duced. Much  labour  has  been  bestowed  upon  the  sec- 
tion on  stairs,  in  which  the  subject  of  hand-railing  is 
presented,  in  many  respects,  in  a  new,  and  it  is  hoped, 
more  practical  form  than  in  previous  treatises  on  that 
subject. 

The  author  has  endeavoured  to  present  a  fund  of  use- 
ful information  to  the  American  house-carpenter  that 
would  enable  him  to  excel  in  his  vocation ;  how  far  he 
has  been  successful  in  that  object,  the  book  itself  must 
determine. 

New  York,  Oct.  15,  1844 


FIFTH  EDITION. 

SFXCE  the  first  edition  of  this  work  was  published,  I 
have  received  numerous  testimonials  of  its  excellent 
practical  value,  from  the  very  best  sources,  viz.  from  the 
workmen  themselves  who  have  used  it,  and  who  have 
profited  by  it  As  a  convenient  manual  for  reference  in 
reepect  to  every  question  relating  either  to  the  simpler. 
operations  of  Carpentry  or  the  more  intricate  and 


PREFACE. 


abstruse  problems  of  Geometry,  those  who  have  tiled 
it  assure  nie  that  they  have  been  greatly  assisted  iu  using 
it.  And,  indeed,  to  the  true  workman,  there  is,  in  the 
study  of  the  subjects  of  which  this  volume  treats,  a  con- 
tinual source  of  profitable  and  pleasurable  interest. 
Gentlemen,  in  numerous  instances,  have  placed  it  in  the 
hands  of  their  sons,  who  have  manifested  a  taste  fop 
practical  studies ;  and  have,  also  procured  it  for  the  use 
of  the  workmen  upon  their  estates,  as  a  guide  in  their 
mechanical  operations.  I  was  not,  then,  mistaken  in  my 
impressions,  that  a  work  of  this  kind  was  wanted ;  and 
this  evidence  of  its  usefulness  rewards  me  in  a  measure 
for  the  pains  taken  in  its  preparation. 

New  York,  Oct.  1,  1852. 


SEYEOTH  EDITION. 

IT  is  now  thirteen  years  since  the  first  edition  of  tht 
American  House  Carpenter  was  published.  The  attempt 
to  furnish  the  recipients  of  this  book  vith  a  fund  of 
useful  information  in  a  compact  and  accessible  form,  has 
been  so  far  successful  that  the  sixth  edition  was  exhausted 
nearly  a  year  ago.  At  that  time  it  was  determined, 
before  issuing  another  edition,  to  make  a  thorough 
revision  of  the  work.  The  time  occupied  in  this  labour 
has  been  unexpectedly  prolonged  by  at  least  six  months, 
and  this  has  resulted  from  various  causes,  but  more 
especially  from  the  absorbing  nature  of  my  professional 
duties.  A  large  portion  of  the  work  has  been  rewritten. 


VI  I' EFFACE. 

about  130  pages  of  iiew  matter  introduced  and  many 
new  cuts  inserted. 

The  most  important  additions  to  the  work  will  be 
found  in  the  section  on  Framing  or  Construction.  Here 
will  be  found,  now  first  published,  the  results  of  experi- 
ments on  such  building  materials  as  are  in  common  use 
in  this  country,  and  an  extended  series  of  rules  for  the 
application  of  this  experimental  knowledge  to  the  prac- 
tical purposes  of  building.  Some  of  the  rules  are  new, 
while  others  heretofore  in  use  have  been  simplified. 
This  section  has  been  much  improved,  and  it  is  hoped 
that  it  will  be  of  service,  not  only  to  the  house  carpenter 
but  also  to  the  architect  and  civil  engineer. 

In  preparing  the  original  work,  a  desire  to  state  the 
subjects  treated  of  in  terms  suited  to  the  comprehension 
of  all  classes  of  workmen,  precluded  the  use  of  algebra- 
ical symbols  and  formulae.  In  this  edition,  however,  it 
has  been  deemed  best  to  introduce  them  wherever  they 
would  contribute  to  the  clearer  elucidation  of  the  sub- 
ject ;  but  care  has  been  taken  to  state  them  in  a  simple 
form  at  first,  and  so  to  explain  the  symbols  as  they  are 
introduced  that  those  heretofore  uninstructed  in  regard 
to  them,  may  comprehend  what  little  is  here  exhibited, 
and  at  the  same  time  be  induced  to  pursue  the  study  more 
fully  in  works  more  strictly  mathematical.  But  for  those 
who  may  not  succeed  in  comprehending  the  algebraical 
formulae,  it  may  be  stated  that  all  the  practical  deductions 
derived  from  them  are  written  out  in  words  at  length, 
so  as  to  be  fully  understood  without  their  assistance. 

*»  York,  KfL  1,  1857.  ^  G'  ^ 


TABLE  OF  CONTEXTS. 


FirrBODUCTiON.— Directions  for  Drawing 1-14 


SECTION  I.-PRACTICAL  GEOMETRY 

Definitions    .                 .        .  15-70 

Problems  on  Lines  and  Angles      ....*...  71-80 

Problems  on  the  Circle 81-92 

Problems  on  Polygons 93-106 

Problems  on  Proportions 107-110 

Problems  on  the  Conic  Sections 111-128 

Demonstrations. — Definitions,  Axioms,  Ac. 130-139 

Demonstrations. — Propositions  and  Corollaries 140-167 

SECTION  IL-ARCHITECTURE. 

i 

History 168-181 

Styles. — Origin,  Definitions,  Proportions 182-196 

Grecian  Orders. — Doric,  Ionic  and  Corinthian 197-211 

Roman  Orders. — Doric,  Ionic,  Corinthian  and  Composite        .        .        .  212-215 

Egyptian  Style 216,217 

Buildings  generally 218-222 

Plans  and  Elevation  for  a  City  Dwelling 223,  224 

Principles  of  Architecture.— Requisites  in  a  Building    ....  225-229 

Principles  of  Construction. — The  Foundations,  Column  ....  230-233 

Principles  of  Construction.— The  Wall,  Lintel,  Arch      ....  233-235 

Principles  of  Construction,— The  Yault,  Dome,  Roof     ....  236-238 


TABLE   OF  CONTENTS. 
SECTION  I1I.-MOULD1NGS,  COB5ICE8,  «ko. 


ABIB 


Mouldings-Elements,  Examples  239-25C 

Cornices.— Designs 

Cornices—Problems •        -      252-256 


SECTION  IV.— FRAMING,  OK  CONSTEUCTION. 

First  Principles.— Laws  of  Pressure 257-282 

Kesistance  of  Materials.— Strength,  Stiffness 283-286 

Resistance  to  Compression. — Various  kinds 287-290 

Results  of  Experiments  on  American  Materials,  Tables  L,  IL        .        .  291-293 

Practical  Rules  and  Examples 294-305 

Resistance  to  Tension 306 

Results  of  Experiments  on  American  Materials,  Table  III.    .  307,  308 

Practical  Rules  and  Examples 309-316 

Resistance  to  Cross  Strains.— Strength,  Stiffness 317-319 

Resistance  to  Deflection.— Stiffness,  Formulae 320-322 

Practical  Rules  and  Examples 323-326 

Table  IV.— Weight  on  Beams,  Formulae 326 

Practical  Rules  and  Examples 327-329 

Table  V. — Dimensions  of  Beams,  Formulae 329 

Resistance  to  Rupture —Strength 331 

Results  of  Experiments  on  American  Materials,  Table  VL   .        .        .  331 

Table  VII.— Safe  "Weight  on  Beams,  Formulas 333 

Practical  Rule*  and  Examples 333 

Table  VIII.— Dimensions  of  Beams,  Formulae 334 

Practical  Rules  and  Fxamples 334 

Systems  of  Framing,  Simplicity  of  Designs 335 

Floors.— Various,  Cross-furring,  Reduction  of  Formulae         .        .        .  336,  337 

Practical  Rules  and  Examples 338-344 

Bridging-strips,  Girders,  Precautions 345-349 

Partitions.— Examples,  Load  on  Partitions,  Ac. 350-353 

Roofs.— Stability,  Inclination 354^  355 

Load.— Roofing,  Truss,  Ceiling,  Wind, -Snow 356-358 

Strains.— Vertical,  Oblique,  Horizontal 359-368 

iMhUnie  of  the  Material  in  Rafter  and  Tie-beam        .  368 

Dimensions.— Rafter,  Braces,  Tie-beam,  Iron  Rods        ....  370-374 

Practical  Rules  and  Examples      ....  37 r  333 

Tible  IX.— Weight  of  Roofs,  per  Foot          ...  37€ 


TABLE   OF   CONTENTS.  IX 

ART* 

Examples  of  Roofs  • 384-3S6 

Problems  for  Hip-rafter 387-388 

Domes.— Examples,  Area  of  Ribs         ....                .  389-391 

Problems  in  Domes .  392-398 

Bridges.— Examples .  399-401 

Rules  for  Dimensions .  401-405 

Abutments  and  Piers 406,  407 

Stone  Bridges,  Centreing 408-417 

Joints  in  Timberwork .  418-427 

*ron  Work.— Pins,  Nails,  Bolts,  Straps          ....                .  428 

Iron-Girders.— Cast  Girder,  Bow-string,  Brick  Arch       .        .                .  429-435 

Practical  Rules  and  Examples .  431-435 

SECTION  V.— DOORS,  WINDOWS,  &a 

Doors. — Dimensions,  Proportions,  Examples          ....  436-441 

Windows. — Form,  Size,  Arrangement,  Problems 442-448 


SECTION  VL-STAIBS, 

Principles,  Pitch  Board 449-456 

Platform  Stairs,  Cylinders,  Rail,  Face  Mould         ....  457^163 

Winding  Stairs,  Falling  Mould,  Face  Mould,  Joints       .        .        .  469-476 

Elucidation  of  Butt  Joint 477 

Quarter-circle  Stairs.— Falling  Mould,  Face  Mould        .        .        .  478-480 

Face  Mould.— Elucidation 481 

Face  Moulds.— Applied  to  Plank,  Bevils,  Ac.  48^-48-1 

Face  Moulds.— Another  method   ......                 .  485-488 

Scrolls,  Rule,  Falling  and  Face  Moulds,  Newel  Cap      .        .        .  489-498 

SECTION  VII.— SHADOWS. 

Shadows  on  Mouldings,  Curves,  Inclinations,  &c.  .  '     .        .        .       .      499-5^2 

Shadows.— Reflected  Light  ..........  523 


APPENDIX. 

MM 

Algebraical  Signs .  3 

Trigonometrical  Terms         ......       0  .        .         6 


JC  TiBLE  OF  CONTENTS. 

PAOl 

Glossary  of  Architectural  Terms .  7 

Table*  of  Squares,  Cubes  and  Roots .13 

Rules  for  Reduction  of  Decimals 27 

Table  of  Areas  and  Circumferences  of  Circle*       ....  29 

Table  of  Capacity  of  Wells,  Cisterns,  4<x 33 

Table  of  Areas  of  Polygons,  Ac,   ....:..  E4 

Table  of  Weights  of  Material* .10 


INTRODUCTION. 


ART.  1. — A  knowledge  of  the  properties  and  principles  of  lines 
c  an  best  be  acquired  by  practice.  Although  the  various  problems 
throughout  this  work  may  be  understood  by  inspection,  yet  they 
\rill  be  impressed  upon  the  mind  with  much  greater  force,  if  they 
are  actually  performed  with  pencil  and  paper  by  the  student. 
Science  is  acquired  by  study — art  by  practice  :  he.  therefore,  who 
would  have  any  thing  more  than  a  theoretical,  (which  must  of 
necessity  be  a  superficial,)  knowledge  of  Carpentry,  will  attend 
to  the  following  directions,  provide  himself  with  the  articles  here 
specified,  and  perform  all  the  operations  described  in  the  follow- 
ing pages.  Many  of  the  problems  may  appear,  at  the  first  read- 
ing, somewhat  confused  and  intricate  ;  but  by  making  one  line 
at  a  time,  according  to  the  explanations,  the  student  will  not 
only  succeed  in  copying  the  figures  correctly,  but  by  ordinary 
attention  will  learn  the  principles  upon  which  they  are  based, 
and  thus  be  able  to  make  them  available  in  any  unexpected  case 
to  which  they  may  apply. 

2. — The  following  articles  are  necessary  for  drawing,  viz  :  a 
drawing-board,  paper,  drawing-pins  or  mouth-glue,  a  sponge,  a 
T-square,  a  set-square,  two  straight-edges,  or  flat  rulers,  a  lead 
pencil,  a  piece  of  india-rubber,  a  cake  of  india-ink,  a  set  of  draw- 
ing-instruments, and  a  scale  of  equal  parts. 

3. — The  size  of  the  drawing-board  must  be  regulated  accord- 
ing to  the  size  of  the  drawings  which  are  to  be  made  upon  k. 
Yet  for  ordinary  practice,  in  learning  to  draw,  a  board  about  15 
1 


2  AMERICAN    HOUSE    CARPENTER. 

by  20  inches,  and  one  inch  thick,  will  be  found  large  enough, 
and  more  convenient  than  a  larger  one.  This  board  should  be 
well-seasoned,  perfectly  square  at  the  corners,  and  without 
clamps  on  the  ends.  A  board  is  better  Without  clamps,  becauss 
the  little  service  they  are  supposed  to  render  by  preventing  the 
board  from  warping,  is  overbalanced  by  the  consideration  that 
the  shrinking  of  the  panel  leaves  the  ends  of  the  clamps  project- 
ing beyond  the  edge  of  the  board,  and  thus  interfering  with  the 
proper  working  of  the  stock  of  the  T-square.  When  the  stuft 
is  well-seasoned,  the  warping  of  the  board  will  be  but  trifling  ; 
and  by  exposing  the  rounding  side  to  the  fire,  or  to  the  sun,  it 
may  be  brought  back  to  its  proper  shape. 

4. — For  mere  line  drawings,  it  is  unnecessary  to  use  the  lest 
drawing-paper ;  and  since,  where  much  is  used  the  expense  will 
be  considerable,  it  is  desirable  for  economy  to  procure  paper 
of  as  low  a  price  as  will  be  suitable  for  the  purpose.  The  best 
paper  is  made  in  England  and  marked  "  Whatman."  This  is 
a  hand-made  paper.  There  is  also  a  machine-made  paper  at 
about  half-price,  and  the  Manilla  paper,  of  various  tints  of  rus- 
set color,  is  still  less  in  price.  These  papers  are  of  the  various 
sizes  needed,  and  are  quite  sufficient  for  ordinary  drawings. 

5.— A  drawing-pin  is  a  small  brass  button,  having  a  steel  pin 
projecting  from  the  under  side.  By  having  one  of  these  at  each 
corner,  the  paper  can  be  fixed  to  the  board  ;  but  this  can  be  done 
in  a  much  better  manner  with  moutk-ghie.  The  pins  will  pre- 
vent the  paper  from  changing  its  position  on  the  board ;  but, 
more  than  this,  the  glue  keeps  the  paper  perfectly  tight  and 
smooth,  thus  making  it  so  much  the  more  pleasant  to  work  on. 

To  attach  the  paper  with  mouth-glue,  lay  it  with  the  bottom 
side  up,  on  the  board ;  and  with  a  straight-edge  and  penknife, 
cut  off  the  rough  and  uneven  edge.  With  a  sponge  moderately 
wet,  nib  all  the  surface  of  the  paper,  except  a  strip  around  the 
edge  about  half  an  inch  wide.  As  soon  as  the  glistening  of  the 
water  disappears,  turn  the  sheet  over,  and  place  it  upon  the 


INTRODUCTION.  3 

board  just  where  you  wish  it  glued.  Commence  upon  one  of 
the  longest  sides,  and  proceed  thus :  lay  a  flat  ruler  upon  the 
paper,  parallel  to  the  edge,  and  within  a  quarter  of  an  inch  of  it 
With  a  knife,  or  any  thing  similar,  turn  up  the  edge  of  the  papei 
against  the  edge  of  the  ruler,  and  put  one  end  of  the  cake  oi 
mouth-glue  between  your  lips  to  dampen  it.  Then  holding  it 
upright,  rub  it  against  and  along  the  entire  edge  of  the  paper 
that  is  turned  up  against  the  roller,  bearing  moderately  against 
the  edge  of  the  ruler,  which  must  be  held  firmly  with  the  left 
hand.  Moisten  the  glue  as  often  as  it  becomes  dry,  until  a 
sufficiency  of  it  is  rubbed  on  the  edge  of  the  paper.  Take 
away  the  ruler,  restore  the  turned-up  edge  to  the  level  of  the 
board,  and  lay  upon  it  a  strip  of  pretty  stiff  paper.  By  rubbing 
upon  this,  not  very  hard  but  pretty  rapidly,  with  the  thumb  nail 
of  the  right  hand,  so  as  to  cause  a  gentle  friction,  and  heat  to  be 
imparted  to  the  glue  that  is  on  the  edge  of  the  paper,  you  will 
make  it  adhere  to  the  board.  The  other  edges  in  succession 
must  be  treated  in  the  same  manner. 

Some  short  distances  along  one  or  more  of  the  edges,  may 
afterwards  be  found  loose :  if  so,  the  glue  must  again  be  applied, 
and  the  paper  rubbed  until  it  adheres.  The  board  must  then  be 
laid  away  in  a  warm  or  dry  place ;  and  in  a  short  time,  the  sur- 
face of  the  paper  will  be  drawn  out,  perfectly  tight  and  smooth, 
and  ready  for  use.  The  paper  dries  best  when  the  board  is  laid 
level.  When  the  drawing  is  finished,  lay  a  straight-edge  upon 
the  paper,  and  cut  it  from  the  board,  leaving  the  glued  strip  still 
attached.  This  may  afterwards  be  taken  off  by  wetting  it  freely 
with  the  sponge ;  which  will  soak  the  glue,  and  loosen  the 
paper.  Do  this  as  soon  as  the  drawing  is  taken  off,  in  order  that 
the  board  rnay  be  dry  when  it  is  wanted  for  use  again.  Care 
must  be  taken  that,  in  applying  the  glue,  the  edge  of  the  paper 
does  not  become  damper  than  the  rest :  if  it  should,  the  paper 
must  be  laid  aside  to  dry,  (to  use  at  another  time,)  and  anothei 
sheet  be  used  in  its  place. 


4  AMERICAN    HOUSE    CARPENTER. 

Sometimes,  especially  when  the  drawing  board  is  new,  the 
paper  will  not  stick  very  readily ;  but  by  persevering,  this  diffi- 
culty may  be  overcome.  In  the  place  of  the  mouth-glue,  a 
strong  solution  of  gum-arabic  may  be  used,  and  on  some 
accounts  is  to  be  preferred ;  for  the  edges  of  the  paper  need  not 
be  kept  dry,  and  it  adheres  more  readily.  Dissolve  the  gum  in 
a  sufficiency  of  warm  water  to  make  it  of  the  consistency  of 
linseed  oil.  It  must  be  applied  to  the  paper  with  a  brush,  when 
the  edge  is  turned  up  against  the  ruler,  as  was  described  for  the 
mouth-glue.  If  two  drawing-boards  are  used,  one  may  be  in  use 
while  the  other  is  laid  away  to  dry  ;  and  as  they  may  be  cheaply 
made,  it  is  advisable  to  have  two.  The  drawing-board  having 
a  frame  around  it,  commonly  called  a  panel-board,  may  afford 
rather  more  facility  in  attaching  the  paper  when  this  is  of  the 
size  to  suit ;  yet  it  has  objections  which  overbalance  that  con 
sideration. 

6  — A  T-square  of  mahogany,  at  once  simple  in  its  construc- 
tion, and  affording  all  necessary  service,  may  be  thus  made. 
Let  the  stock  or  handle  be  seven  inches  long,  two  and  a  quarter 
inches  wide,  and  three-eighths  of  an  inch  thick:  the  blade, 
twenty  inches  long,  (exclusive  of  the  stock,)  two  inches  wide, 
and  one-eighth  of  an  inch  thick.  In  joining  the  blade  to  the 
stock,  a  very  firm  and  simple  joint  may  be  made  by  dovetailing 
it— as  shown  at  Fisr.  1. 


INTRODUCTION.  B 

7. — The  set -square  is  in  the  form  of  a  right-angled  triangle  ; 
and  is  commonly  made  of  mahogany,  one-eighth  of  an  inch  in 
thickness.  The  size  that  is  most  convenient  for  general  use,  is 
six  inches  and  three  inches  respectively  for  the  sides  which  con- 
tain  the  right  angle ;  although  a  particular  length  for  the  sides  is 
by  no  means  necessary.  Care  should  be  taken  to  have  the  square 
corner  exactly  true.  This,  as  also  the  T-square  and  rulers, 
should  have  a  hole  bored  through  them,  by  which  to  hang  them 
upon  a  nail  when  not  in  use. 

8. — One  of  the  rulers  may  be  about  twenty  inches  long,  and 
the  other  six  inches.  The  pencil  ought  to  be  hard  enough  to 
retain  a  fine  point,  and  yet  not  so  hard  as  to  leave  ineffaceable 
marks.  It  should  be  used  lightly,  so  that  the  extra  marks  that 
are  not  needed  when  the  drawing  is  inked,  may  be  easily  rubbed 
off  with  the  rubber.  The  best  kind  of  india-ink  is  that  which 
will  easily  rub  off  upon  the  plate  ;  and,  when  the  cake  is  rub- 
bed against  the  teeth,  will  be  free  from  grit. 

9. — The  drawing-instruments  may  be  purchased  of  mathe- 
matical instrument  makers  at  various  prices  :  from  one  to  one 
hundred  dollars  a  set.  In  choosing  a  set,  remember  that  the 
lowest  price  articles  are  not  always  the  cheapest.  A  set,  com- 
prising a  sufficient  number  of  instruments  for  ordinaiy  use,  well 
made  and  fitted  in  a  mahogany  box,  may  be  purchased  of  the 
mathematical  instrument-makers  in  New  York  for  four  or  five 
dollars.  But  for  permanent  use  those  which  come  at  ten  or 
twelve  dollars  will  be  found  to  be  the  best. 

10. — The  best  scale  of  equal  parts  for  carpenters'  use,  is  one 
that  has  one-eighth,  three-sixteenths,  one-fourth,  three-eighths, 
one-half,  five-eighths,  three-fourths,  and  seven-eighths  of  an 
inch,  and'  one  inch,  severally  divided  into  twelfths,  instead  ot 
being  divided,  as  they  usually  are,  into  tenths.  By  this,  if  it  be 
required  to  proportion  a  drawing  so  that  every  foot  of  the  object 
represented  will  upon  the  paper  measure  one-fourth  of  an  inch, 
use  that  part  of  the  scale  which  is  divided  into  one-fourths  of  an 


6  AMERICAN    HOUSE-CARPENTER. 

inch  taking  for  every  foot  one  of  those  divisions,  and  for  every 
inch  one  of  the  subdivisions  into  twelfths  ;  and  proceed  in  like 
manner  in  proportioning  a  drawing  to  any  of  the  other  divisions 
of  the  scale.  An  instrument  in  the  form  of  a  semi-circle,  called  a 
protractor,  and  used  for  laying  down  and  measuring  angles,  is 
of  much  service  to  surveyors,  but  not  much  to  carpenters. 

11. — In  drawing  parallel  lines,  when  they  are  to  be  parallel 
to  either  side  of  the  board,  use  the  T-square ;  but  when  it  is 
required  to  draw  lines  parallel  to  a  line  which  is  drawn  in  a 
direction  oblique  to  either  side  of  the  board,  the  set-square  must 
be  used.  Let  a  b,  (Fig.  2,)  be  a  line,  parallel  to  which  it  is 


Fig  a. 


desired  to  draw  one  or  more  lines.  Place  any  edge,  as  c  d,  ol 
the  set-square  even  with  said  line ;  then  place  the  ruler,  g  /«, 
against  one  of  the  other  sides,  as  c  e,  and  hold  it  firmly ;  slide 
the  set-square  along  the  edge  of  the  ruler  as  far  as  it  is  desired, 
as  at/;  and  a  line  drawn  by  the  edge,  »'/,  will  be  parallel  to  a  b. 
12.— To  draw  a  line,  as  k  I  (Fig.  3,)  perpendicular  to  another, 
as  a  6,  set  the  shortest  edge  of  the  set-square  at  the  line,  a  b; 
place  the  ruler  against  the  longest  side,  (the  hypothenuse  of  the 
right-angled  triangle ;)  hold  the  ruler  firmly,  and  slide  the  set- 
square  along  until  the  side,  e  d  touches  the  point,  k  ;  then  the 
line,  I  k,  drawn  by  it,  will  be  perpendicular  to  a  b.  In  like 


INTRODUCTION. 


manner  the  drawing  of  other  problems  may  be  facilitated,  as  will 
be  discovered  in  using  the  instruments. 


Pig.  3. 


13. — In  drawing  a  problem,  proceed,  with  the  pencil  sharpened 
to  a  point,  to  lay  down  the  several  lines  until  the  whole  figure  is 
completed ;  observing  to  let  the  lines  cross  each  other  at  the 
several  angles,  instead  of  merely  meeting.  By  this,  the  length 
of  every  line  will  be  clearly  defined.  With  a  drop  or  two  of 
vater,  rub  one  end  of  the  cake  of  ink  upon  a  plate  or  saucer, 
until  a  sufficiency  adheres  to  it.  Be  careful  to  dry  the  cake  of 
ink  ;  because  if  it  is  left  wet,  it  will  crack  and  crumble  in  pieces. 
With  an  inferior  camel's-hair  pencil,  add  a  little  water  to  the 
ink  that  was  rubbed  on  the  plate,  and  mix  it  well.  It  should  be 
diluted  sufficiently  to  flow  freely  from  the  pen,  and  yet  be  thick 
enough  to  make  a  black  line.  With  the  hair  pencil,  place  a 
little  of  the  ink  between  the  nibs  of  the  drawing-pen,  ancf  screw 
the  nibs  together  until  the  pen  makes  a  fine  line.  Beginning 
with  the  curved  lines,  proceed  to  ink,  all  the  lines  of  the  figure  ; 
being  careful  now  to  make  every  line  of  its  requisite  length.  If 
they  are  a  trifle  too  short  or  too  long,  the  drawing  will  have  a 
ragged  appearance;  and  this  is  opposed  to  that  neatness  and 
accuracy  which  is  indispensable  to  a  good  drawing.  When  the 
ink  is  diy,  efface  the  pencil-marks  with  the  india-rubber.  If 


8  AMERICAN    HOUSE-CARPENTER. 

the  pencil  is  used  lightly,  they  will  all  rub  off,  leaving  those 
lines  only  that  were  inked.    . 

14. — In  problems,  all  auxiliary  lines  are  drawn  light ;  while 
the  lines  given  and  those  sought,  in  order  to  be  distinguished  at 
a  glance,  are  made  much  heavier.  The  heavy  lines  are  made 
so,  by  passing  over  them  a  second  time,  having  the  nibs  of  the 
pen  separated  far  enough  to  make  the  lines  as  heavy  as  desired. 
If  the  heavy  lines  are  made  before  the  drawing  is  cleaned  with 
the  rubber,  they  will  not  appear  so  black  and  neat ;  because  the 
india-rubber  takes  away  part  of  the  ink.  If  the  drawing  is  a 
ground-plan  or  elevation  of  a  house,  the  shade-lines,  as  they  are 
termed,  should  not  be  put  in  until  the  drawing  is  shaded ;  as 
there  is  danger  of  the  heavy  lines  spreading,  when  the  brush,  in 
shading  or  coloring,  passes  over  them.  If  the  lines  are  inked 
with  common  writing-ink,  they  will,  however  fine  they  may  be 
made,  be  subject  to  the  same  evil ;  for  which  reason,  india-ink 
is  the  only  kind  to  be  used. 


THE 

AMERICAN    HOUSE-CARPENTER, 


SECTION    I.— PRACTICAL    GEOMETRY. 


DEFINITIONS. 

15.  -  Geometry  treats  of  the  properties  of  magnitudes, 

16. — A  point  has  neither  length,  breadth,  nor  thickness. 

17.  —A  line  has  length  only. 

18. — Superficies  has  length  and  breadth  only. 

19. — A  plane  is  a  surface,  perfectly  straight  and  even  in  every 
direction  ;  as  the  face  of  a  panel  when  not  warped  nor  winding. 

20. — A  solid  has  length,  breadth  and  thickness. 

21. — A  right,  or  straight,  line  is  the  shortest  that  can  be 
drawn  between  two  points.  . 

22. — Parallel  lines  are  equi-distant  throughout  their  length. 

23. — An  angle  is  the  inclination  of  two  lines  towards  one 
another.  (Fig.  4.) 


Fif.  4.  Fig.  5.  Fig.  & 

2 


10  AMERICAS!    HOUSE-CARPENTER. 

24. — A  right  angle  has  one  line  perpendicular  to  the  othei. 
(Fig.  5.) 

25. — An  oblique  angle  is  either  greater  or  less  than  a  right 
angle.  (Fig.  4  and  6.) 

26. — An  acute  angle  is  less  than  a  right  angle.     (Fig-  4.) 

27. — An  obtuse  angle  is  greater  than  a  right  angle.     (Fig.  6.) 

When  an  angle  is  denoted  hy  three  letters,  the  middle  one,  in 
the  order  they  stand,  denotes  the  angular  point,  and  the  other 
two  the  sides  containing  the  angle  ;  thus,  let  a  b  c,  (Fig.  4,)  be 
the  angle,  then  b  will  be  the  angular  point,  and  a  b  and  b  c  will 
De  the  two  sides  containing  that  angle. 

28. — A  triangle  is  a  superficies  having  three  sides  and  angles, 
(Fig.  7,  8,  9  and  10.) 


2- — An   equi-lateral  triangle    has    its    three   sides   equal. 
(IV- 7.) 

30. — An  isosceles  triangle  has  only  two  sides  equal.     (Fig.  8.) 
Si. — A  scalene  triangle  has  all  its  sides  unequal.     (Fig.  9) 


Fig.  10. 


32.— A  right-angled  triangle  has  one  right  angle.     (Fig.  10.) 
33.— An   acute-angled    triangle    has    all   its   angles   acute. 
(Fig.  7  and  8.) 

34.— An    obtuse-angled    triangle    has    one    obtuse    angle, 
(Fig.  9.) 

35.— A  quadrangle  has  four  sides  and  four  angles.     (Fig.  11 
to  16.) 


PRACTICAL    GEOMETRY. 


11 


Fig.  11. 


Fif.  12. 


36. — A  parallelogram  is  a  quadrangle  having  its  opposite 
sides  parallel.  (Fig.  11  to  14.) 

37. — A  rectangle  is  a  parallelogram,  its  angles  being  right 
angles.  (Fig.  11  and  12.) 

38. — A  square  is  a  rectangle  having  equal  sides.     (Fig.  11 .) 

39. — A  rhombus  is  an  equi-lateral  parallelogram  having  ob- 
lique angles.  (Fig.  13.) 


Fig.  13. 


Fig.  14. 


40. — A  rhomboid  is  a  parallelogram  having  oblique  angles. 
(Fig.  14.) 

41. — A  trapezoid  is  a  quadrangle  having  only  two  of  its  sides 
parallel.  (Fig.  15.) 


Fig.  15. 


Fig.  16. 


42. — A  trapezium  is  a  quadrangle  which  has  no  two  of  its 
sides  parallel.  (Fig.  16.) 

43. — A  polygon  is  a  figure  bounded  by  right  lines. 

44. — A  regular  polygon  has  its  sides  and  angles  equal. 

4.5. — An  irregular  polygon  has  its  sides  and  angles  unequal. 

46. — A  trigon  is  a  polygon  of  three  sides,  (Fig.  7  to  10 ,) 
a  tetragon  has  four  sides,  (Fig.  11  to  16 :)  a  pentagon  has 


|2  AMERICAN    HOUSE-CARPENTER. 

five,  (Fig.  17  ;)  a  hexagon  six,  (Fig.  18 ;)  a  heptagon  seven, 
(Fig.  19 ;)  an  octagon  eight,  (Fig.  20 ;)  a  nonagon  nine  ;  a 
decagon  ten  ;  an  undecagon  eleven  ;  and  a  dodecagon  twelve 
sides. 


Fif.  17. 


Fig.  1&. 


Fig.  19. 


Fig.  20. 


47. A  circ/e  is  a  figure  bounded  by  a  curved  line,  called  the 

circumference  ;  which  is  every  where  equi-distant  from  a  cer- 
tain point  within,  called  its  centre. 

The  circumference  is  also  called  the  periphery,  and  sometimes 
the  circle. 

48. — The  radius  of  a  circle  is  a  right  line  drawn  from  the 
centre  to  any  point  in  the  circumference,  (a  6,  Fig.  21.) 

All  the  radii  of  a  circle  are  equal. 


Fn.Ul. 


49. — The  diameter  is  a  right  line  passing  through  the  centre, 
and  terminating  at  two  opposite  points  in  the  circumference. 
Hence  it  is  twice  the  length  of  the  radius,  (c  d,  Fig.  21.) 

50. — An  arc  of  a  circle  is  a  part  of  the  circumference,  (c  b  or 
bed,  Fig.  21.) 

51. — A  chord  is  a  right  line  joining  the  extremities  of  an  arc. 
(Id,  Fig.  21.) 


PRACTICAL    GEOMETRY.  13 

52. — A  segment  is  any  part  of  a  circle  bounded  by  an  arc  and 
its  chord.  (A,  Fig.  21.) 

53. — A  sector  is  any  part  of  a  circle  bounded  by  an  arc  and 
two  radii,  drawn  to  its  extremities.  (B,  Fig.  21.) 

54. — A  quadrant,  or  quarter  of  a  circle,  is  a  sector  having  a 
quarter  of  the  circumference  for  its  arc.  (C,  Fig.  21.) 

55. — A  tangent  is  a  right  line,  which  in  passing  a  curve, 
touches,  without  cutting  it.  (/  g-,  Fig.  21.) 

56. — A  cone  is  a  solid  figure  standing  upon  a  circular  base 
diminishing  in  straight  lines  to  a  point  at  the  top,  called  its 
vortex.  (Fig.  22.) 


Fig.  22. 


57. — The  axis  of  a  cone  is  a  right  line  passing  through  it,  from 
the  vertex  to  the  centre  of  the  circle  at  the  base. 

58. — An  ellipsis  is  described  if  a  cone  be  cut  by  a  plane,  not 
parallel  to  its  base,  passing  quite  through  the  curved  surface. 
(a  b,  Fig.  23.) 

59. — A  parabola  is  described  if  a  cone  be  cut  by  a  plane, 
parallel  to  a  plane  touching  the  curved  surface,  (c  d,  Fig:  23 — 
c  d  being  parallel  to  f  g.} 

60. — An  hyperbola  is  described  if  a  cone  be  cut  by  a  plane, 
parallel  to  any  plane  within  the  cone  that  passes  through  its 
vertex,  (e  h,  Fig.  23.) 

61. — Foci  are  the  points  at  which  the  pins  are  placed  in  de- 
scribing an  ellipse.  (See  Art.  115,  and/,  /,  Fig.  24.) 


u 


AMERICAN    HOUSE-CARPENTER. 


62. — The  transverse  axis  is  the  longest  diameter  of  the 
ellipsis,  (a  6,  Fig.  24.) 

63. — The  conjugate  axis  is  the  shortest  diameter  of  the 
ellipsis  ;  and  is,  therefore,  at  right  angles  to  the  transverse  axis, 
(e  d,  Fig.  24.) 

64. — The  parameter  is  a  right  line  passing  through  the  focus 
of  an  ellipsis,  at  right  angles  to  the  transverse  axis,  and.  termina- 
ted by  the  curve,  (g  h  and  g  t,  Fig.  24.) 

65. — A  diameter  of  an  ellipsis  is  any  right  line  passing 
through  the  centre,  and  terminated  by  the  curve,  (k'l,  or  m  n, 
Fig.  24.) 

66. — A  diameter  is  conjugate  to  another  when  it  is  parallel  to 
a  tangent  drawn  at  the  extremity  of  that  other — thus,  the  diame- 
ter, m  H,  (Fig.  24,)  being  parallel  to  the  tangent,  o  p,  is  therefore 
conjugate  to  the  diameter,  k  I. 

67. — A  double  ordinate  is  any  right  line,  crossing  a  diameter 
of  an  ellipsis,  and  drawn  parallel  to  a  tangent  at  the  extremity  of 
that  diameter,  (i  t,  Fig.  24.) 

68. — A  cylinder  is  a  solid  generated  by  the  revolution  of  a 
right-angled  parallelogram,  or  rectangle,  about  one  of  its  sides ; 
and  consequently  the  ends  of  the  cylinder  are  equal  circles. 
(Fig.  25.) 


PRACTICAL     GEOMETRY. 


15 


Fig.  25. 


Fig.  26. 


69. — The  axis  of  a  cylinder  is  a  right  line  passing  through  it, 
from  the  centres  of  the  two  circles  which  form  the  ends. 

70. — A  segment  of  a  cylinder  is  comprehended  under  three 
planes,  and  the  curved  surface  of  the  cylinder.  Two  of  these 
are  segments  of  circles  :  the  other  plane  is  a  parallelogram,  called 
by  way  of  distinction,  the  plane  of  the  segment.  The  circular 
segments  are  called,  the  ends  of  the  cylinder.  (Fig.  26.) 

JV".  B. — For  Algebraical  Signs,  Trigonometrical  Terms,  &c., 
see  Appendix. 


PROBLEMS. 


RfGHT      LINES     AND     ANGLES. 

71. — To  bisect  a  line.    Upon  the  ^ndr  of  the  line,  a  6,  (Fig. 
27,)  as  centres,  with  any  distance  for  radius  greater  than  hall 


d 

Fig.  27. 

a  6j  describe  arcs  cutting  each  other  in  c  and  d  ;  draw  the  line, 
c  if,  and  the  point,  e,  where  it  cuts  a  b,  will  be  the  middle  of  the 
line,  a  b. 

In  practice,  a  line  is  generally  divided  with  the  compasses,  or 
dividers  ;  but  this  problem  is  useful  where  it  is  desired  to.  draw, 
at  the  middle  of  another  line,  one  at  right  angles  to  it.  (See 
Art.  8*5.) 


Fig.  28. 


72.—  To  erect  a  perpendicular.     From  the  point,  a,  (Fig.  28,) 


PRACTICAL    GEOMETRY. 


17 


set  oft  any  distance,  as  a  6,  and  the  same  distance  from  a  to  c  , 
upon  c,  as  a  centre,  with  any  distance  for  radius  greater  than  c  a, 
describe  an  arc  at  d  ;  upon  6,  with  the  same  radius,  describe 
another  at  d ;  join  d  and  a,  and  the  line,  d  a,  will  be  the  per- 
pendicular required. 

This,  and  the  three  following  problems,  are  more  easily  per- 
formed by  the  use  of  the  set-square — (see  Art.  12.)  Yet  they 
are  useful  when  the  operation  is  so  large  that  a  set-square  cannot 
be  used. 


73._  To  let  fall  a  perpendicular.  Let  a,  (Fig.  29.)  be  the 
point,  above  the  line,  b  c,  from  which  the  perpendicular  is  re- 
quired to  fall.  Upon  a,  with  any  radius  greater  than  a  d,  de- 
scribe an  arc,  cutting  b  c  at  e  and/;  upon  the  points,  e  and/ 
with  any  radius  greater  than  e  d,  describe  arcs,  cutting  each 
other  at  g  ;  join  a  and  g,  and  the  line,  a  d,  will  be  the  perpen- 
dicular required. 


Fig.  30. 

74. —  To  erect  a  perpendicular  at  the  end  of  a  line.  Let  a, 
(Fig.  30,)  at  the  end  of  the  line,  c  a,  be  the  point  at  which  the 
perpendicular  is  to  be  erected.  Take  any  point,  as  6,  above  the 
line,  c  a,  and  with  the  radius,  6  a,  describe  the  arc,  d  a  c. 
through  d  and  6,  draw  the  line,  d  e  ;  join  e  and  a,  then  e  a  will 
be  the  perpendicular  required. 


18 


AMERICAN    HOUSE-CARPENTER. 


The  principle  here  made  use  of,  is  a  very  important  one  ;  and 
is  applied  in  many  other  cases— (see  Art.  81,  b.  and  Art.  84. 
For  proof  of  its  correctness,  see  Art.  156.) 


Fig.  31. 

74,  a. — A  second  method.  Let  6,  (Fig.  31,)  at  the  end  of  the 
lino,  a  6,  be  the  point  at  which  it  is  required  to  erect  a  perpen- 
dicular. Upon  6,  with  any  radius  less  than  b  a,  describe  the  arc, 
c  e  d  ;  upon  c,  with  the  same  radius,  describe  the  small  arc  at  e, 
and  upon  e,  another  at  d  ;  upon  e  and  d,  with  the  same  or  any 
other  radius  greater  than  half  e  c?,  describe  arcs  intersecting  at/; 
join /and  6,  and  the  line,/ 6,  will  be  the  perpendicular  required. 
This  method  of  erecting  a  perpendicular  and  that  of  the  fol- 
lowing article,  depend  for  accuracy  upon  the  fact  that  the  side 
of  a  hexagon  is  equal  to  the  radius  of  the  circumscribing  circle. 


d 

Fig.  32. 


74,  b.— A  third  method.  Let  b,  (Fig.  32,)  be  the  given  point 
at  which  it  is  required  to  erect  a  perpendicular.  Upon  b}  with  any 
radius  less  than  b  a,  describe  the  quadrant,  d  ef;  upon  d.  with 
the  same  radius,  describe  an  arc  at  e,  and  upon  e,  another  at  c , 


PRACTICAL    GEOMETRY.  19 

through  d  and  e,  draw  d  c,  cutting  the  arc  in  c;  jcin  c  and  6, 
then  c  b  will  be  the  perpendicular  required. 

This  problem  can  be  solved  by  the  six,  eight  and  ten  rule 
as  it  is  called  ;  which  is  founded  upon  the  same  principle  as 
the  problems  at  Art.  103,  104 ;  and  is  applied  as  follows.  Let 
a  d,  (Fig.  30,)  equal  eight,  and  a  e,  six  ;  then,  if  d  e  equals  ten. 
the  angle,  e  a  d,  is  a  right  angle.  Because  the  square  of  six 
and  that  of  eight,  added  together,  equal  the  square  of  ten,  thus  : 
6  x  6  =  36,  and  8  x  8  =  64 ;  36  +  64  =  100,  and  10  x  10  = 
100.  Any  sizes,  taken  in  th'e  same  proportion,  as  six,  eight  and 
ten,  will  produce  the  same  effect :  as  3,  4  and  5,  or  12,  16  and 
20.  (See  Art.  103.) 

By  the  process  shown  at  Fig.  30,  the  end  of  a  board  may  be 
squared  without  a  carpenters'-square.  All  that  is  necessary  is  a 
pair  of  compasses  and  a  ruler.  Let  c  a  be  the  edge  of  the  board, 
and  a  the  point  at  which  it  is  required  to  be  squared.  Take  the 
point,  b,  as  near  as  possible  at  an  angle  of  forty-five  degrees,  or  on 
a  mitre-line,  from  a,  and  at  about  the  middle  of  the  board.  This 
is  not  necessary  to  the  working  of  the  problem,  nor  does  it  affect 
its  accuracy,  but  the  result  is  more  easily  obtained.  Stretch  the 
compasses  from  b  to  a,  and  then  bring  the  leg  at  a  around  to  d  ; 
draw  a  line  from  d,  through  b,  out  indefinitely ;  take  the  dis- 
tance, d  b,  and  place  it  from  b  to  e  ;  join  e  and  a  ;  then  e  a  will 
be  at  right  angles  to  c  a.  In  squaring  the  foundation  of  a  build- 
ing, or  laying-out  a  garden,  a  rod  and  chalk-line  may  be  used 
instead  of  compasses  and  ruler. 

75. —  To  let  fall  a  perpendicular  near  the  end  of  a  line. 
Let  e,  (Fig.  30,)  be  the  point  above  the  line,  c  a,  from  which  the 
perpendicular  is  required  to  fall.  From  e,  draw  any  line,  as  e  d, 
obliquely  to  the  line,  c  a  ;  bisect  e  d  at  b  ;  upon  6,  with  the 
radius,  b  e,  describe  the  arc,  e  a  d  ;  *join  e  and  a  ;  then  e  a  will 
be  the  perpendicular  required. 


d 
Fig.  33. 


76. —  To  make  an  angle,  (as  e  df,  Fig.  33,)  equal  to  a  given 
angle,  (as  b  a  c.)  From  the  angular  point,  a,  with  any  radius 
describe  the  arc,  be;  and  with  .the  same  radius,  on  the  line,  d  e% 


20  AMERICAN    HOUSE-CARPENTER. 

and  from  the  point,  rf,  describe  the  arc,/£-;  take  the  distance, 
b  c,  and  upon  g,  describe  the  small  arc  at/;  join /and  d  ;  and 
the  angle,  e  df,  will  be  equal  to  the  angle,  b  a  c. 

If  the  given  line  upon  which  the  angle  is  to  be  made,  is  situa- 
ted parallel  to  the  similar  line  of  the  given  angle,  this  may  be 
performed  more  readily  with  the  set-square.  (See  Art.  11.) 


Fig.  34. 


77.— To  bisect  an  angle.  Let  a  b  c,  (Fig.  34,)  be  the  angle 
to  be  bisected.  Upon  6,  with  any  radius,  describe  the  arc,  a  c  ; 
upon  a  and  c,  with  a  radius  greater  than  half  a  c,  describe  arcs 
cutting  each  other  at  d  ;  join  b  and  d  ;  and  b  d  will  bisect  the 
angle,  a  6  c,  as  was  required. 

This  problem  is  frequently  made  use  of  in  solving  other  pro- 
blems ;  it  should  therefore  be  well  impressed  upon  the  memory. 


Fig.  35. 

78. —  To  trisect  a  right  ang'c.  Upon  a,  (Fig.  35,)  with  any 
radius,  describe  the  arc",  b  c  ;  upim  b  and  c,  with  the  same  radius, 
describe  arcs  cutting  the  arc,  b  c,  at  d  and  e  ;  from  d  and  e,  draw 
lines  to  a,  and  they  will  trisect  the  angle  as  was  required. 

The  truth  of  this  is  made  evident  by  the  following  operation. 
Divide  a  circle  into  quadrants  :  also,  take  the  radius  in  the  divi- 
ders, and  space  off  the  circumference.  This  will  divide  the 
circumference  into  just  six  parts.  A  semi-circumference,  there- 


PRACTICAL     GEOMETRY.  21 

tore,  is  equal  to  three,  and  a  quadrant  to.  one  and  a  half  of  those 
parts.  The  radius,  therefore,  is  equal  to  f  of  a  quadrant ;  and 
this  is  equal  to  a  right  angle. 


Fig.  36. 

79. —  Through  a  given  point,  to  draw  a  line  parallel  to  a 
given  line.  Let  a,  (Fig.  36,)  be  the  given  point,  and  b  c  the 
given  line.  Upon  any  point,  as  d,  in  the  line,  b  c,  with  the 
radius,  d  a,  describe  the  arc,  a  c;  upon  a,  with  the  same  radius, 
describe  the  arc,  d  e  ;  make  d  e  equal  to  a  c  ;  through  e  and  a 
draw  the  line,  e  a  ;  which  will  be  the  line  required. 

This  is  upon  the  same  principle  as  Art.  76. 


.s  e  J  g  h  b 

Fig.  37 

80. —  To  divide  a  given  line  into  any  number  of  equal  parts. 
Let  a  b)  (Fig.  37,)  be  the  given  line,  and  5  the  number  of  parts. 
Draw  a  c,  at  any  angle  to  a  b  ;  on  a  c,  from  a,  set  off  5  equal 
parts  of  any  length,  as  at  1,  2,  3,  4  and  c  ;  join  c  and  b  ;  through 
the  points,  1,  2,  3  and  4,  draw  1  e,  2f,  3  g  and  4  h,  parallel  to 
c  b  ;  which  will  divide  the  line,  a  b,  as  was  required. 

The  lines,  a  b  and  a  c,  are  divided  in  the  same  proportion. 
(See  Art.  1Q9.) 

T'flE     CIRCLE. 

81. —  To  find  the  centre  of  a  circle.     Draw  any  chord,  as  a  b 


AMERICAN    HOUSE-CARPENTER. 


(Fig.  38,)  and  bisect  it  with  the  perpendicular,  c  d  ;  hisect  c  d 
with  the  line,  e/,  as  at  g  ;  then  g  is  the  centre  as  was  required. 


81,  a. — A  second  method.  Upon  any  two  points  in  the  cir- 
cumference nearly  opposite,  as  a  and  b.  (Fig-  39,)  describe  arcs 
cutting  each  other  at  c  and  d :  take  any  other  two  joints,  as  e 
and/,  and  describe  arcs  intersecting  as  at  g  and  h  ;  join  g  and  h, 
and  r,  and  d  ;  the  intersection,  0,  is  the  centre. 

This  is  upon  the  same  principle  as  Art.  85. 


81,  b.— A  third  method.     Draw  any  chord,  as  a  b.  (Fig.  40,) 


PRACTICAL    GEOMETRY. 


and  from  the  point,  a,  draw  a  e,  at  right  angles  to  a  b  ;  join 
c  and  b  ;  bisect  c  b  at  d  —  which  will  be  the  centre  of  the  circle. 

If  a  circle  be  not  too  large  for  the  purpose,  its  centre  may  very 
readily  be  ascertained  by  the  help  of  a  carpenters'-square,  thus  : 
app'  y  the  corner  of  the  square  to  any  point  in  the  circumference, 
as  at  a  ;  by  the  edges  of  the  square,  (which  the  lines,  a  b  and 
a  c,  represent,)  draw  lines  cuttirig  the  circle,  as  at  b  and  c  ;  join 
b  and  t  ;  then  if  b  c  is  bisected,  as  at  d,  the  point,  d,  will  be  the 
centre.  (See  Art.  156.) 


Fig.  41. 


82. — At  a  given  point  in  a  circle,  to  draw  a  tangent  thereto, 
Let  c,  (Fig.  41,)  be  the  given  point,  and  b  the  centre  of  the  cir- 
cle.    Join  a  and  b  ;  through  the  point,  o,  and  at  right  angles  to 
a  6,  draw  c  d;  then  c  d  is  the  tangent  required. 
d a 


Fig.  42 

B3. —  The  same,  without  making  use  of  the  centre  of  the 
circle.  Let  a,  (Fig.  42,)  be  the  given  point.  From  a,  set  on* 
any  distance  to  6,  and  the  same  from  b  to  c ;  join  a  and  c , 
upon  a,  with  a  b  for  radius,  describe  the  arc,  d  b  e  ;  make  d  b 
equal  to  be;  through  a  arid  d,  draw  a  line ;  this  will  be  the 
tangent  required. 

The  correctness  of  this  method  depends  upon  the  fact  that 
the  angle  formed  by  a  chord  and  tangent  is  equal  to  any 


24  AMERICAN   nOUSK-CABPEXTER. 

inscribed  angle  in  the  opposite  segment  of  the  circle,  (Art, 
163 ;)  a  b  being  the  chord,  and  I  c  a  the  angle  in  the  opposite 
segment  of  the  circle.  Now,  the  angles  d  a  I  and  b  c  a  are 
equal,  because  the  angles  dab  and  b  a  c  are,  by  construction, 
equal ;  and  the  angles  b  a  c  and  b  c  a  are  equal,  because  the 
triangle  a  b  c  is  an  isosceles  triangle,  having  its  two  sides,  a  b 
and  b  c,  by  construction  equal ;  therefore  the  angles  dab  and 
b  c  a  are  equal. 


84.— A  circle  and  a  tangent  given,  to  find  the  point  of  con- 
tact. From  any  point,  as  a,  (Fig.  43,)  in  the  tangent,  b  c,  draw 
a  line  to  the  centre  d;  bisect  a  d  at  e;  upon  e,  with  the  radins, 
e  a,  describe  the  arc,  afd;f  is  the  point  of  contact  required. 

If/ and  d  were  joined,  the  line  would  form  right  ano-les  with 
the  tangent,  b  c.  (See  Art.  156.) 


S3.— Through  any  three  points  not  in  a  straight  line,  to  draw 
a  circle.  Let  a,  b  and  c,  (Fig.  44,)  be  the  three  given  points. 
Upon  a  and  J,  with  any  radiug  greater  than  half  a  b,  describe 


PRACTICAL   GEOMETRY.  25 

arcs  intersecting  a!;  d  and  e ,'  upon  b  and  c,  witn  any  radius 
greater  than  half  b  c,  describe  arcs  intersecting  at  /  and  g  • 
through  d  and  e,  draw  a  right  line,  also  another  through  f  and 
g  /  upon  the  intersection,  h,  with  the  radius,  h  a,  describe  the 
circle,  a  J  c,  and  it  will  be  the  one  required. 


86. — Three  points  not  in  a  straight  line  being  given,  to  find 
a  fourth  that  shall,  with  the  three,  lie  in  the  circumference  of  a, 
circle.  Let  a  b  c,  (Fig.  45,)  be  the  given  points.  Connect 
them  with  right  lines,  forming  the  triangle,  a  c  b  j  bisect  the 
angle,  c  b  a,  (Art.  77,)  with  the  line  b  d  /  also  bisect  c  a  in  e, 
and  erect  e  d,  perpendicular  to  a  c,  cutting  J  d  in  d;  then  d  is 
the  fourth  point  required. 

A  fifth  point  may  be  found,  as  at/",  by  assuming  a,  d  and  b, 
as  the  three  given  points,  and  proceeding  as  before.  So,  also, 
any  number  of  points  may  be  found ;  simply  by  using  any  three 
already  found.  This  problem  will  be  serviceable  in  obtaining 
short  pieces  of  very  flat  sweeps.  (See  Art.  397.) 

The  proof  of  the  correctness  of  this  method  is  found  in  the 
fact  that  equal  chords  subtend  equal  angles,  (Art.  162.)  Join 
d  and  c;  then  since  a  e  and  e  c  are,  by  construction,  equal, 
therefore  the  chords  a  d  and  d  c  are  equal ;  hence  the  angles 
they  subtend,  d  b  a  and  d  b  c,  are  equal.  So  likewise  chords 
drawn  from  a  to/",  and  from  f  to  d,  are  equal,  and  subtend  the 
equal  angles,  d  bf  and  f  b  a.  Additional  points,  beyond  a  or 
5,  may  be  obtained  on  the  same  principle.  To  obtain  a  point 
beyond  a,  on  b,  as  a  centre,  describe  with  any  radius  the  arc 
i  o  n,  make  o  n  equal  to  o  i;  through  5  and  n  draw  b  g  ;  on  a  as 


26  AMERICAN    HOUSE-CARPENTER. 

centre  and  with  af  for  radius,  describe  the  arc,  cutting  g  I  at 
0,  then  g  is  the  point  sought. 


87. — To  describe  a  segment  of  a  circle  ty  a  set-triangle.  Let 
a  b,  (Fig.  46,)  be  the  chord,  and  c  d  the  height  of  the  segment. 
Secure  two  straight-edges,  or  rulers,  in  the  position,  c  e  and  cf, 
by  nailing  them  together  at  c,  and  affixing  a  brace  from  e  to 
//  put  in  pins  at  a  and  b  ;  move  the  angular  point,  c,  in  the 
direction,  a  c  b  /  keeping  the  edges  of  the  triangle  hard  against 
the  pins,  a  and  o  /  a  pencil  held  at  c  will  describe  the  arc,  a  c  o. 

A  curve  described  by  this  process  is  accurately  circular,  and 
is  not  a  mere  approximation  to  a  circular  arc,  as  some  may 
suppose.  This  method  produces  a  circular  curve,  because  all 
inscribed  angles  on  one  side  of  a  chord  line  are  equal.  (Art. 
161.)  To  obtain  the  radius  from  a  chord  and  its  versed  sine, 
see  Art.  165. 

If  the  angle  formed  by  the  rulers  at  c  be  a  right  angle,  the 
segment  described  will  be  a  semi-circle.  This  problem  is  use- 
in^  in  describing  centres  for  brick  arches,  when  they  are  re- 
quired to  be  rather  flat.  Also,  for  the  head  hanging-stile  of  a 
window-frame,  where  a  brick  arch,  instead  of  a  stone  lintel,  is 
to  be  placed  over  it. 

87  a. — To  find  the  radius  of  an  arc  of  a  circle  when  the 
chord  and  versed  sine  are  given.  The  radius  is  equal  to  the 
sum  of  the  squares  of  half  the  chord  and  of  the  versed  sine, 
divided  by  twice  the  versed  sine.  This  is  expressed,  algebraic- 
ally, thus— r=-~—^  where  r  is  the  radius,  c  the  chord,  and  v 
the  versed  sine.  (Art.  165.) 

Example.— In  a  given  arc  of  a  circle,  a  chord  of  12  feet  has 


PRACTICAL   GEOilETEY.  27 

tne  rise  at  the  middle,  or  the  versed  sine,  equal  to  2  feet,  what 

is  the  radius  ? 

Half  the  chord  equals  6,  the  square  of  6  is,  6  X  6  =  36 
The  square  of  the  versed  sine  is,  2  x  2  =  4 

Their  sum  equals,  40 

Twice  the  versed  sine  equals  4,  and  40  divided  by  4  equals  10 
Therefore  the  radius,  in  this  case,  is  10  feet.  This  result  it 
shown  in  less  space  and  more  neatly  by^  using  the  above  alge- 
braical formula.  For  the  letters,  substituting  their  value,  the 

(e.y   I   -y*  (^r)s  -f-  2a 

formula  r  —  -^~- —  becomes  r  =  —^ — — ,  and  performing 

2v  2x2' 

the  arithmetical  operations  here  indicated  equals 
62  +  2'  _  36  +  4  _  40 

~T~      ~T~  -  T  = 

87  b. — To  find  the  versed  sine  of  an  arc  of  a  circle  when  the 
radius  and  chord  are  given.  The  versed  sine  is  equal  to  the 
radius,  less  the  square  root  of  the  difference  of  the  squares  of 
the  radius  and  half  chord  :  expressed  algebraically  thus — v  =  T 
—  y  r*  —  (!)",  where  r  is  the  radius,  v  the  versed  sine,  and  o 
the  chord.  (Art.  158.) 

Example. — In  an  arc  of  a  circle  whose  radius  is  75  feet, 
what  is  the  versed  sine  to  a  chord  of  120  feet  ?  By  the  table 
in  the  Appendix  it  will  .be  seen  that — 

The  square  of  the  radius,  75,  equals,        .         .     5625 
The  square  of  half  the  chord,  60,  equals,          .     3600 

The  difference  is, '.     2025 


The  square  root  of  this  is,        .        .        .        .45 
leducted  from  the  radius,        ...        75 


The  sqi 
This  de 


The  remainder  is  the  versed  sine,  •=  30 

This  is  expressed  by  the  formula  thus — 
=  75  -  V75a-  (-^)3=75  —  V  5625~^3600  =  75  —  45  =  3C 


3         d 
Fig.  47. 


28  AMERICAN   HOUSE-CAlCrENTER. 

88.— To  deso-ibe  the  segment  of  a  circle  ly  intersection  of 
lines.  Let  a  5,  (Fig.  47,)  be  the  chord,  and  c  d  the  height  of 
the  segment.  Through  c,  draw  e /,  parallel  toal;  draw  I  f 
at  right  angles  to  c  b  ;  make  c  e  equal  to  c  f;  draw  a  g  and 
b  A,  at  right  angles  to  a  I ;  divide  c  e,  cf,  d  a,db,a  g,  and 
b  A,  each  into  a  like  number  of  equal  parts,  as  four  ;  draw  the 
lines,  1  1,  2  2,  &c.,  and  from  the  points,  o,  o  and  o,  draw  lines 
to  c  ;  at  the  intersection  of  these  lines,  trace  the  curve,  a  c  b, 
which  will  be  the  segment  required. 

In  very  large  work,  or  in  laying  out  ornamented  gardens, 
&c.,  this  will  be  found  useful ;  and  where  the  centre  of  the 
proposed  arc  of  a  circle  is  inaccessible  it  will  be  invaluable. 
(To  trace  the  curve,  see  note  at  Art.  117.) 

The  lines  e  a,  c  d  and  / 1,  would,  were  they  extended,  meet 
in  a  point,  and  that  point  would  be  in  the  opposite  side  of  the 
circumference  of  the  circle  of  which  a  c  b  is  a  segment,  The 
lines  1  1,  2  2,  3  3,  would  likewise,  if  extended,  meet  in  the 
same  point.  The  line,  c  d,  if  extended  to  the  opposite  side  of 
the  circle,  would  become  a  diameter.  The  line,/ 5,  forms,  by 
construction,  a  right  angle  with  b  c,  and  hence  the  extension  of 
f  b  would  also  form  a  right  angle  with  b  c,  on  the  opposite  side 
of  b  c  /  and  this  right  angle  would  be  the  inscribed  angle  in 
the  semicircle ;  and  since  this  is  required  to  be  a  right  angle, 
(Art.  156,)  therefore  the  construction  thus  far  is  correct,  and  it 
will  be  found  likewise  that  at  each  point  in  the  curve  formed 
by  the  intersection  of  the  radiating  lines,  these  intersecting 
lines  are  at  right  angles. 


Fig.  47  a. 


88  a. — Points  in  the  circumference  of  a  circle  may  be  ob- 
tained arithmetically,  and  positively  accurate,  by  the  calcula- 
tion of  ordinateS)  or  the  parallel  lines,  0  1,  0  2,  0  3,  0  4.  (Fig. 


PRACTICAL   GEOMETRY.        •  29 

4:7  «.)  These  ordinates  are  drawn  at  right  angles  to  the  chord 
line,  a  5,  and  they  may  be  drawn  at  any  distance  apart,  eithei 
equally  distant  or  unequally,  and  there  may  be  as  many  of 
them  as  is  desirable ;  the  more  there  are  the  more  points  in 
the  curve  will  be  obtained.  If  they  are  located  in  pairs, 
equally  distant  from  the  versed  sine,  c  d,  calculation  need  be 
made  only  for  those  on  one  side  of  c  d,  as  those  on  the  opposite 
side  will  be  of  equal  lengths,  respectively  ;  for  example,  0  1,  on 
fhe  left-hand  side  of  c  d,  is  equal  to  0  1  on  the  right-hand  side, 
0  2  on  the  right  equals  0  2  on  the  left,  and  in  like  manner  for 
the  others. 

The  length  of  any  ordinate  is  equal  to  the  square  root  of 
the  difference  of  the  squares  of  the  radius  and  abscissa,  less 
the  difference  between  the  radius  and  versed  sine.  (Art.  166.) 
The  abscissa  being  the  distance  from  the  foot  of  the  versed  sine 
to  the  foot  of  the  ordinate.  Algebraically,  y  —  -v/?'2—  #2  — 
(;.  __  <y)?  where  y  is  put  to  represent  the  ordinate ;  x,  the  ab- 
scissa ;  -y,  the  versed  sine  ;  and  r,  the  radius. 

Example. — An  arc  of  a  circle  has  its  chord,  a  5,  (Fig.  47" «,) 
100  feet  long,  and  its  versed  sine,  c  d,  5  feet.  It  is  required  to 
ascertain  the  length  of  ordinates  for  a  sufficient  number  of 
points  through  which  to  describe  the  curve.  To  this  end  it  is 
requisite,  first,  to  ascertain  the  radius.  This  is  readily  done  in 

accordance  with  Art.  87  a.     For,  H>L — -,  becomes  \      „    = 

2i  V  a   X   0 

252-5  =  radius.  Having  the  radius,  the  curve  might  at  once 
be  described  without  the  ordinate  points,  but  for  the  impracti- 
cability that  usually  occurs,  in  large,  flat  segments  of  the  circle, 
of  getting  a  location  for  the  centre  ;  the  centre  usually  being 
inaccessible.  The  ordinates  are,  therefore,  to  be  calculated. 
In  Fig.  47  a  the  ordinates  are  located  equidistant,  and  are  10 
feet  apart.  It  will  only  be  requisite,  therefore,  to  calculate 
those  on  one  side  of  the  versed  sine,  c  d.  Fondie  first  ordinate, 
0  1,  the  formula,  y  =  Vr'J  —  x*  —  (r  —  v)  becomes 
y  =  -v^52'5'  -  IP"5  -  (252-5  -  5). 

-  V63756-25  —  100  -  247'5. 

=          252-3019  -  247-5. 

=  4-8019  =  the  first  ordinate,  0  1. 


30  AMI.RICAN   HOUSH -CARPENTEE. 

For  tue  bccoml — • 

y  =  v'252'5'  -  '20*  -(252'5  -  5). 
=      251-7066  —  247-5. 

4-2066  =  the  second  ordinate,  02. 
For  the  third— 

y  =  1/252-5'  -  30*  —  247'5. 
=      250-7115  —  247-5. 
=         3-2115  =  the  third  ordinate,  03. 

For  the  fourth — 

y  =  -v/252'5'  -  403  -  247'5. 
=      249-3115  —  247-5. 

1-8115  =  the  fourth  ordinate,  04. 

The  results  here  obtained  are  in  feet  and  decimals  of  a  foot. 
To  reduce  these  to  feet,  inches,  and  eighths  of  an  inch,  proceed 
as  at  Reduction  of  Decimals  in  the  Appendix.  If  the  two-feet 
rule,  used  by  carpenters  and  others,  were  decimally  divided, 
there  would  be  no  necessity  of  this  reduction,  and  it  is  to  be 
hoped  that  the  rule  will  yet  be  thus  divided,  as  such  a  reform 
would  much  lessen  the  labor  of  computations,  and  insure  more 
accurate  measurements. 

Versed  sine,  c  d,  =  ft.  5-0  -  ft.  5'0  inches. 

.Ordinates,      01,=       4'8019       =       4'9f  inches  nearly. 

"  02,=       4-2066       =       4-2  i  inches  nearly. 

"  03,=       3-2115       =       3-2$  inches  .nearly. 

"  04,=       1-8115       =       1-91  inches  nearly. 


Fig.  48. 

89- — In  a  given  angle,  to  describe  a  tanged  curve.  Let  a  5  c, 
(Fig.  48,)  be  the  given  angle,  and  1  in  the  line,  a  &,  and  5  in 
the  line,  b  c,  the  termination  of  the  curve.  Divide  1  I  and  I  5 


into  a  like  number  of  equal  parts,  as  at  1,  2,  3,  4  and  5  ;  join  1 
and  1,  2  and  2,  3  and  3,  &a;  and  a  regular  curve  will  be 
formed  that  will  be  tangical  to  the  line,  a  5,  at  the  point,  1,  and 
to  b  o  at  5. 

This  is  of  mnch  use  ir.  stair-building,  in  easing  the  angles 
Formed  between  the  wall-string  and  the  base  of  the  hall,  also 


PRACTICAL   GEOMETRY. 


31 


between  the  front  string  and  level  facia,  and  in  many  other 
instances.  The  curve  is  not  circular,  but  of  the  form  of  the 
parabola,  (Fig.  93 ;)  yet  in  large  angles  the  difference  is  not 
perceptible.  This  problem  can  be  applied  to  describing  the 


c 
Fig.  49. 


curve  for  door  heads,  window-heads,  &c.,  to  rather  better  ad- 
vantage than  Art.  87.  For  instance,  let  a  £,  (Fig.  49,}  be  the 
width  of  the  opening,  and  c  d  the  height,  of  the  arc.  Extend  c 
d,  and  make  d  e  equal  to  c  d  ;  join  a  and  e,  also  e  and  b  /  and 
proceed  as  directed  above. 


90. — To  describe  a  circle  within  any  g^ven  t/riangle,  so  that 
the  sides  of  the  triangle  shall  be  iangical.  Let  a  b  c,  (Fig.  50,) 
be  the  given  triangle.  Bisect  the  angles  a  and  b,  according  to 
Art.  77 ;  upon  d,  the  point  of  intersection  of  the  bisecting  lines, 
with  the  radius,  d  e,  describe  the  required  circle. 


32  AMERICAN   HOUSE-CARPENTER. 

91.— About  a  given  circle,  to  describe  an  equi-lateral  tri- 
angU.  Let  a  d  be,  (Fig.  51,)  be  the  given  circle.  Draw  the 
diameter,  c  d;  upon  d,  with  the  radius  of  the  given  circle, 
describe  the  arc,  a  el;  join  a  and  I ;  draw/  g,  at  right  angles 
to  d  c  ;  make/  c  and  c  g,  each  equal  to  a  I ;  from  /  through 
a,  draw /A,  also  from  0,  through  b,  draw  0  A;  then/0-  A.  will 
be  the  triangle  required. 


'    p 


92.  —  To  find  a  right  line  nearly  equal  to  the  circumference 
of  a  circle.  Let  a  b  c  d,  (Fig.  52,)  be  the  given  circle.  Draw 
the  diameter,  a  c  ;  on  this  erect  an  equi-lateral  triangle,  a  e  c, 
according  to  Art.  93  ;  draw  g  /,  parallel  to  a  c  /  extend  e  c  to 
/,  also  e  a  to  g  /  then  g  f  will  be  nearly  the  length  of  tha 
semi-circle,  a  d  c  /  and  twice  gf  will  nearly  equal  the  circum- 
ference of  the  circle,  a  b  c  d,  as  was  required. 

Lines  drawn  from  e,  through  any  points  in  the  circle,  as  0,  o 
and  0,  to  p,p  and  ^p,  will  divide  g  f  in  the  same  way  as  the 
semi-circle,  a  d  c,  is  divided.  So,  any  portion  of  a  circle  may 
be  transferred  to  a  straight  line.  This  is  a  very  useful  pro- 
blem, and  should  be  well  studied  ;  as  it  is  frequently  used"  to 
solve  problems  on  stairs,  domes,  &c. 

.92,  a.  —  Another  method.  Let  a  bf  c,  (Fig.  53,)  be  the  given 
circle.  Draw  the  diameter,  a  c  ;  from  d,  the  centre,  and  at  right 
angles  to  a  c,  draw  d  1;  join  I  and  c;  bisect  I  c  at  e;  from  d, 
through  ^,  draw  df;  then  ef  added  to  three  times  the  diameter, 
•will  equal  the  circumference  of  the  circle  sufficiently  near  for 


PKACTICAL   GEOMETKT.  33 


Fig.  53. 


many  uses.  The  result  is  a  trifle  too  large,  If  the  circumfer- 
erence  found  by  this  rule,  be  divided  by  648*22,  the  quotient 
will  be  the  excess.  Deduct  this  excess,  and  the  remainder 
will  be  the  true  circumference.  This  problem  is  rather  more 
curious  than  useful,  as  it  is  less  labor  to  perform  the  operation 
arithmetically:  simply  multiplying  the  given  diameter  by 
3-1416,  or  where  a  great  degree  of  accuracy  is  needed  by 
31415926. 

POLYGONS,   &0. 


b 
Fig.  64. 


93. —  Upon  a  given  line  to  construct  an  equi-lateral  triangle. 
Let  a  J,  (Fig.  54,)  be  the  given  line.  Upon  a  and  5,  with  a  o 
for  radius,  describe  arcs,  intersecting  at  c  ;  join  a  and  c,  also  c 
and  1)  •  then  a  c  5  will  be  the  triangle  required. 

94. — To  describe  an  equi-lateral  rectangle,  or  square.    Let 

a  5,  (Fig.  55,)  be  the  length  of  a  side  of  the  proposed  square. 

Upon  a  and  5,  with  a  &  for  radius,  describe  the  arcs  a  d  and 

b  c  j  bisect  the  arc,  a  <?,  in  f  •  upon  0,  with  e  f  for  radius,  de- 

5 


AMERICAN   HOCSE-CAKPENTER. 


Fig.  55. 


scribe  the  arc,  cfd;  join  a  and  c,  c  and  d,  d  and  I ;  then  a  c 
d  b  will  be  the  square  required. 


95. —  Within  a  given  circle,  to  inscribe  an  equi-lateral  tri* 
angle,  hexagon  or  dodecagon.  Let  a  o  c  d,  (Fig.  56,)  be  the 
given  circle.  Draw  the  diameter,  o  d;  upon  5,  with  the 
radius  of  the  given  circle,  describe  the  arc,  a  e  c  /  join  a  and 
c,  also  a  and  d,  and  c  and  d — and  the  triangle  is  completed. 
For  the  hexagon:  from  a,  also  from  c,  through  e,  draw  the 
lines,  a  f  and  c  g  ;  join  a  and  b, 1  and  c,  c  and  /,  &c.,  and  the 
hexagon  is  completed.  The  dodecagon  may  be  formed  by 
bisecting  the  sides  of  the  hexagon. 

Each  side  of  a  regular  hexagon  is  exactly  equal  to  the 
radius  of  the  circle  that  circumscribes  the  figure.  For  the 
radius  is  equal  to  a  chord  of  an  arc  of  60  degrees ;  and,  as 
every  circle  is  supposed  to  be  divided  into  360  degrees,  there 
is  just  6  times  60,  or  6  arcs  of  60  degrees,  in  the  wliole  circum- 
ference. A  line  drawn  from  each  angle  of  the  hexagon  to  the 
centre,  (as  in  the  figure,)  divides  it  into  six  equal,  equi-lateral 
triangles. 

96. —  Within  a  square  to  inscribe  an  octagon.    Let  a  o  c  d, 


PRACTICAL    GEOMETRY. 


35 


(Fig.  57,)  be  the  given  square.  Draw  the  diagonals,  a  d  and 
b  c  /  upon  a,  b,  c  and  d,  with  a  e  for  radius,  describe  arcs  cut- 
ting the  sides  of  the  square  at  1,  2,  3,  4,  5,  6,  7  and  8  ;  join  1 
and  2,  3  and  4,  5  and  6,  &c.,  and  the  figure  is  completed. 

In  order  to  eight-square  a  hand-rail,  or  any  piece  that  is  to 
be  afterwards  rounded,  draw  the  diagonals,  a  d  and  b  c,  upon 
the  end  of  it,  after  it  has  been  squared-up.  Set  a  gauge  to  the 
distance,  a  e,  and  run  it  upon  the  whole  length  of  the  stuff, 
from  each  corner  both  ways.  This  will  show  how  much  is  to 
be  chamfered  off,  in  order  to  make  the  piece  octagonal.  (Art. 
159.) 


Fig.  58. 


97.—  Within  a  given  circle  to  inscribe  any  regular  polygon. 
Let  a  I  c  2,  (Fig.  58,  59  and  60,)  be  given  circles.  Draw  the 
diameter,  a  c ;  upon  this,  erect  an  equi-lateral  triangle,  a  e  c, 
according  to  Art.  93 ;  divide  a  c  into  as  many  equal  parts  as 
the .  polygon  is  to  have  sides,  as  at  1,  2,  3,  4,  &c. ;  from  e, 


36  AMERICAN  HOUSE-CARPENTER. 

through  each  even  number,  as  2,  4,  6,  &c.,  draw  lines  cutting 
the  circle  in  the  points,  2,  4,  &c. ;  from  these  points  and  at 
right  angles  to  a  <?,  draw  lines  to  the  opposite  part  of  the  circle  \ 
this  will  give  the  remaining  points  for  the  polygon,  as  £,/,  &c. 
In  forming  a  hexagon,  the  sides  of  the  triangle  erected  upon 
a  c,  (as  at  Fig.  59,).  mark  the  points  5  and  /.  This  method  of 
locating  the  angles  of  a  polygon  is  an  approximation  suffi- 
ciently near  for  many  purposes ;  it  is  based  upon  the  like  prin- 
ciple with  the  method  of  obtaining  a  right  line  nearly  equal  to 
a  circle.  (Art.  92.)  The  method  shown  at  Art.  98  is  accurate. 


Fig'  61.  Fig.  62.  Fig.  63. 

98. —  Upon  a  given  line  to  describe  any  regular  polygon. 
Let  a  J,  (Fig.  61,  62  and  63,)  be  given  lines,  equal  to  a  side  of 
the  required  figure.  From  5,  draw  b  c,  at  right  angles  to  a  o  / 
upon  a  and  I,  with  a  o  for  radius,  describe  the  arcs,  a  c  d  and 
fe  b  £  divide  a  c  into  as  many  equal  parts  as  the  polygon  is  to 
have  sides,  and  extend  those  divisions  from  c  towards  d  j  from 
the  second  point  of  division  counting  from  c  towards  #,  as  3, 
(Fig.  6.1,)  4,  (Fig.  62,)  and  5,  (Fig.  63,)  draw  a  line  to  ~b  ;  take 
the  distance  from  said  point  of  division  to  a,  and  set  it  from  o 
to  e;  join  e  and  a  ;  upon  the  intersection,  0,  with  the  radius, 
o  «,  describe  the  circle  a  f  d  I ;  then  radiating  lines,  drawn 
from  I  through  the  even  numbers  on  the  arc,  a  d,  will  cut  the 
circle  at  the  several  angles  of  the  required  figure. 

In  the  hexagon,  .(Fig.  62,)  the  divisions  on  the  arc,  a  d,  are  not 
necessary ;  for  the  noint,  o,  is  at  the  intersection  of  the  arcs,  a  d 
and/ ft,  the  points,/ and  d,  are  determined  by  the  intersection  of 
those  arcs  with  the  circle,  and  the  points  above,  g  and  /*,  can  be 
found  by  drawing  lines  from  a  and  5,  through  the  centre,  o.  In 
polygons  of  a  greater  number  of  sides  than  the  hexagon,  the  in- 
tersection, o,  comes  above  the  arcs  ;  in  such  case,  therefore,  the 


PRACTICAL   GEOMETRY.  37 

lines  a  e  and  J  5,  (Fig.  63,)  have  to  be  extended  before  they  will 
intersect.  This  method  of  describing  polygons  is  founded  on 
correct  principles,  and  is  therefore  accurate.  In  the  circle  equal 
arcs  subtend  equal  angles,  (Arts.  86  and  162.)  Although  this 
method  is  accurate,  yet  polygons  may  be  described  as  accu- 
rately and  more  simply  in  the  following  manner.  It  will  be 
observed  that  much  of  the  process  in  this  method  is  for  the  pur 
pose  of  ascertaining  the  centre  of  a  circle  that  will  circumscribe 
the  proposed  polygon.  By  reference  to  the  Table  of  Polygons 
in  the  Appendix 'it  will  be  seen  how  this  centre  may  be  obtained 
arithmetically.  This  is  the  Rule. — Multiply  the  given  side  by 
the  tabular  radius  for  polygons  of  a  like  number  of  sides  with 
the  proposed  figure,  and  the  product  will  be  the  radius  of  the 
required  circumscribing  circle.  Divide  this  circle  into  as  many 
equal  parts  as  the  polygon  is  to  have  sides,  connect  the  points  of 
division  by  straight  lines,  and  the  figure  is  complete.  For  exam- 
ple :  It  is  desired  to  describe  a  polygon  of  7  sides,  and  20  inches 
a  side.  The  tabular  radius  is  1*1523824:.  This  multiplied  by 
20,  the  product,  23'04764S  is  the  required  radius  in  inches.  The 
Rules  for  the  Reduction  of  Decimals,  also  in  the  Appendix, 
show  how  to  change  decimals  to  the  fractions  of  a  foot  or  an 
inch.  From  this,  23-047648  is  equal  to  23TV  inches  nearly.  It 
is  not  needed  to  take  all  the  decimals  in  the  table,  three  or  four  of 
them  will  give  a  result  sufficiently  near  for  all  ordinary  practice. 


Fie.  64. 

99. — To  construct  a  triangle  whose  sides  shall  be  severally 
equal  to  three  given  lines.  Let  a,  b  and  <?,  (Fig.  64,)  be  the  given 
lines.  Draw  the  line,  d  e,  and  make  it  equal  to  c  ;  upon  0,  with 
I  for  radius,  describe  an  arc  at  f  •  upon  d,  with  a  for  radius, 
describe  an  arc  intersecting  the  other  at  f  /  join  d  and  f,  also 
f  and  e  /  then  df  e  will  be  the  triangle  required. 


Fig.  65.  Fig.  66. 


38  PRACTICAL     GEOMETRY. 

100. —  To  construct  a  fgure  equal  to  a  g.ven,  right-lined 
figure.  Let  a  b  c  d,(Fig.  65,)  be  the  given  figure.  Make  e/, 
(Fig.  66,)  equal  to  c  d  ;  upon  /,  with  d  a  for  radius,  describe  an 
arc  at  g;  upon  e,  with  c  a  for  radius,  describe  an  arc  intersecting 
the  other  at  g  ;  join  g  and  e  ;  upon  /  and  g,  with  d  b  and  a  b 
for  radius,  describe  arcs  intersecting  at  h  ;  join  g  and  h,  also  h 
and/;  then  Fig.  66  will  everyway  equal  Fig.  65. 

So,  right-lined  figures  of  any  number  of  sides  may  be  copied, 
r>y  first  dividing  them  into  triangles,  and  then  proceeding  as 
above.  The  shape  of  the  floor  of  any  room,  or  of  any  piece  of 
land,  &c.,  may  be  accurately  laid  out  by  this  problem,  at  a  scale 
upon  paper ;  and  the  contents  in  square  feet  be  ascertained  by 
the  next. 


V. \J. 


d 

Fig.  67. 


101. —  To  make  a  parallelogram,  equal  to  a  given  triangle. 
Let  a  b  c,  (Fig.  67,)  be  the  given  triangle.  From  a,  draw  a  d, 
at  right  angles  to  be;  bisect  a  d  in  e  ;  through,  c,  Arawfg, 
parallel  to  b  c  ;  from  b  and  c,  draw  b  f  and  c  g,  parallel  to  d  e  ; 
then  bfg  c  will  be  a  parallelogram  containing  a  surface  exactly 
equal  to  that  of  the  triangle,  a  b  c. 

Unless  the  parallelogram  is  required  to  be  a  rectangle,  the  lines, 
I /and  c  g,  need  not  be  drawn  parallel  to  d  e.  If  a  rhomboid  is 

esired  they  may  be  drawn  at  an  oblique  angle,  provided  they 
ae  parallel  to  one  another.  To  ascertain  the  area  of  a  triangle, 
multiply  the  base,  b  c,  by  half  the  perpendicular  height,  d  a.  In 

>mg  this,  it  matters  not  which  side  is  taken  for  We, 


AMERICAN    HOUSE-CARPENTER. 


•39 


102. — A  parallelogram  being  given,  to  construct  anotkei 
equal  to  it,  and  having  a  side  equal  to  a  given  line.  Let  A 
(Fig.  68,)  be  the  given  parallelogram,  and  B  the  given  line 
Produce  the  sides  of  the  parallelogram,  as  at  a,  b,  c  and  d  ;  make 
e  d  equal  to  B  ;  through  d,  draw  c  /,  parallel  to  g  b  ;  through 
e,  draw  the  diagonal,  c  a ••;  from  a,  draw  a  f,  parallel  to  e  d 
then  C  will  be  equal  to  A.  (See  Art.  144.) 


Fig  69. 

103. —  To  make  a  square  equal  to  two  or  more  given  squares, 
Let  A  and  B,  (Fig.  69,)  be  two  given  squares.  Place  them  sc 
as  to  form  a  right  angle,  as  at  a  ;  join  b  and  c  ;  then  the  square, 
C,  formed  upon  the  line,  b  c,  will  be  equal  in  extent  to  the  squares, 
A  and  B,  added  together.  Again  :  if  a  b,  (Fig.  70,)  be  equal  to 


the  side  of  a  given  square,  c  a,  placed  at  right  angleo  to  a  b,  be  the 
side  of  another  given  square,  and  c  d,  placed  at  right  angles  to 


40  PRACTICAL    GEOMETRY. 

c  6,  be  the  side  of  a  third  given  square ;  then  the  square,  A, 
formed  upon  the  line,  d  6,  will  be  equal  to  the  three  given 
squares.  (See  Art.  157.) 

The  usefulness  and  importance  of  this  problem  are  proverbial. 
To  ascertain  the  length  of  braces  and  of  rafters  in  framing,  the 
length  of  stair-strings,  &c.,  are  some  of  the  purposes  to  which  it 
may  be  applied  in  carpentry.  (See  note  to  Art.  74,  b.}  If  the 
length  of  any  two  sides  of  a  right-angled  triangle  is  known,  that 
of  the  third  can  be  ascertained.  Because  the  square  of  the 
hypothenuse  is  equal  to  the  united  squares  of  the  two  sides  that 
contain  the  right  angle. 

(1.) — The  two  sides  containing  the  right  angle  being  known, 
to  find  the  hypothenuse.  Rule. — Square  each  given  side,  add 
the  squares  together,  and  from  the  product  extract  the  square- 
root  :  this  will  be  the  answer.  For  instance,  suppose  it  were 
required  to  find  the  length  of  a  rafter  for  a  house,  34  feet  wide, — 
the  ridge  of  the  roof  to  be  9  feet  high,  above  the  level  of  the 
wall-plates.  Then  17  feet,  half  of  the  span,  is  one,  and  9  feet, 
the  height,  is  the  other  of  the  sides  that  contain  the  right  angle 
Proceed  as  directed  by  the  rule : 

17  9 

17  9 

119  81  =  square  of  9. 

17  289  =  square  of  17. 

289  -  square  of  17.     370  Product. 

1  )  370  (  19-235  +  =  square-root  of  370  ;  equal  19  feet,  2;-  in. 
1      1  nearly :  which  would  be  the  required 

length  of  the  rafter. 
29 )  270 
9    261 

382) --900 
2     764 

3843 )  13600 
3    11529 

38465)-  207100        (By  reference  to  the  table  of  square-roots 

192325     in  the  Appendix,  the  root  of  almost  any 

number   may  be  found   ready  calculated  ; 

also,  to  change  the  decimals  of  a  foot  to  inches  and  parts,  see 

Rules  for  the  Reduction  of  Decimals  in  the  Appendix.) 


AMERICAN    HOUSE-CARPENTER.  4} 

A  galr  :  suppose  it  be  required,  in  a  frame  building,  to  rind  the 
length  of  a  brace,  having  a  run  of  three  feet  each  way  from  the 
point  of  the  right  angle.  The  length  of  the  sides  containing  the 
right  angle  will  be  each  3  feet :  then,  as  before— 

3 
3 

9  =  square  of  one  side. 
3  times  3  =  9=  -square  of  the  other  side. 

1 8  Product :  the  square-root  of  which  is  4*2426  +  ft., 
or  4  feet,  2  inches  and  |ths.  full. 

(2.) — The  hypothenuse  and  one  side  being- known,  to  find  the 
other  eide.  Rule. — Subtract  the  square  of  the  given  side  from 
the  square  of  the  hypothenuse,  and  the  square-root  of  the  product 
will  be  the  answer.  Suppose  it  were  required  to  ascertain  the 
greatest  perpendicular  height  a  roof  of  a  given  span  may  have, 
when  pieces  of  timber  of  a  given  length  are  to  be  used  as  rafters. 
Let  the  span  be  20  feet,  and  the  rafters  of  3x4  hemlock  joist. 
These  come  about  13  feet  long.  The  known  hypothenuse, 
then,  is  13  feet,  and  the  known  side,  10  feet — that  being  half  the 
span  of  the  building. 

13 
13 

39 
13 

169  =  square  of  hypothenuse. 
lO  times  10  =  100  =  square  of  the  given  side. 

69  Product :  the  square-root  of  which  is  8 
•3066  +  feet,  or  8  feet,  3  inches  and  Mhs.  full.  This  will  be. 
the  greatest  perpendicular  height,  as  required.  Again  :  suppose 
that  in  a  story  of  8  feet,  from  floor  to  floor,  a  step-ladder  is  re- 
quired, the  strings  of  which  are  to  be  of  plank,  12  feet  long  ;  and 
it  is  desirable  to  know  the  greatest  run  such  a  length  of  string 
will  afford.  In  this  case,  the  two  given  sides  are — hypothenuse 
12,  perpendicular  8  feet. 

12  times  12  -=  144  =  square  of  hypothenuse. 
8  times    8  =    64  =  square  of  perpendicular. 

80  Product :  the  square-root  of  which  is  8'9442  -f 
feet,  or  8  feet,  "*!  inches  and  ^ths. — the  answer,  as  required. 
6 


PRACTICAL    GEOMETRY. 


Many  other  cases  might  be  adduced  to  show  the  utility  of  this 
problem.  A  practical  and  ready  method  of  ascertaining  tlid 
length  of  braces,  rafters,  &c.,  when  not  of  a  great  length,  is  to 
apply  a  rule  across  the  carpenters'-square.  Suppose,  for  the 
length  of  a  rafter,  the  base  be  12  feet  and  the  height  7.  Apply 
the  rule  diagonally  on  the  square,  so  that  it  touches  12  inches 
from  the  corner  on  one  side,  and  7  inches  from  the  corner  on  the 
other.  The  number  of  inches  on  the  rule,  which  are  intercepted 
by  the  sides  of  the  square,  13t  nearly,  will  be  the  length  of  the 
rafter  in  feet  ;  viz,  13  feet  and  |-ths  of  a  foot.  -If  the  dimensions 
are  large,  as  30  feet  and  20,  take  the  half  of  each  on  the  sides  of 
the  square,  viz,  15  and  10  inches  ;  then  the  length  in  inches 
across,  will  be  one-half  the  number  of  feet  the  rafter  is  long. 
This  method  is  just  as  accurate  as  the  preceding  ;  but  when 
the  length  of  a  very  long  rafter  is  sought,  it  requires  great  care 
and  precision  to  ascertain  the  fractions.  For  the  least  variation 
on  the  square,  or  in  the  length  taken  on  the  rule,  would  make 
perhaps  several  inches  difference  in  the  length  of  the  rafter. 
For  shorter  dimensions,  however,  the  result  will  be  true  enough. 


104. —  To  make  a  circle  equal  to  two  given  circles.  Let  A 
and  #,  (Fig.  71,)  be  the  given  circles.  In  the  right-angled  tri- 
angle, a  6  c,  make  a  b  equal  to  the  diameter  of  the  circle,  B,  and 
z  b  equal  to  the  diameter  of  the  circle,  A  ;  then  the  hypothenuse. 


Tig.  78. 


AMERICAN    HOUSE-CARPENTER.  ,« 

a  ct  will  be  the  diameter  of  a  circle,  C,  which  will  be  equal  in 
area  to  the  two  circles,  A  and  B,  added  together. 

Any  polygonal  figure,  as  A,  (Fig.  72,)  formed  on  the  hypo- 
thenuse  of  a  right-angled  triangle,  will  be  equal  to  two  similar 
figures,*  as  B  and  C*,  formed  on  the  two  legs  of  the  triangle. 


Fig.  73 

105. —  To  construct  a  square  equal  to  a  given  rectangle. 
Let  A,  (Fig.  73,)  be  the  given  rectangle.  Extend  the  side,  a  6, 
and  make  b  c  equal  to  b  e  ;  bisect  a  c  in/,  and  upon  f,  with  the 
radius,  /  a,  describe  the  semi-circle,  age;  extend  e  b,  till  it 
cuts  the  curve  in  g  ;  then  a  square,  b  g  h  d,  formed  on  the  line, 
b  ff,  will  be  equal  in  area  to  the  rectangle,  A. 


105,  a. — Another  method.     Let  A,  (Fig.  74,)  be  the  given 
rectangle.     Extend  the  side,  a  6,  and  make  a  d  equal  to  a  c, 

*  Similar  figures  are  such  as  have  their  several  angles  respectively  equal,  and  theii 
aiijs  respectively  proportionate. 


44  PRACTICAL    GEOMETRY. 

bisect  a  d  in  e  ;  upon  c,  with  the  radius,  e  a,  describe  the  semi- 
circle, a/  d  ;  extend  £•  b  till  it  cuts  the  curve  in/;  join  a  arid 
/';  then  the  square,  B,  formed  on  the  line,  a/,  will  be  equal  in 
area  to  the  rectangle,  A.  (See  Art.  156  and  157.) 

106. —  To  form  a  square  equal  to  a  given  triangle.  Let  a  6, 
(Fig.  73,)  equal  the  base  of  the  given  triangle,  and  b  e  equal 
half  its  perpendicular  height,  (see  Fig.  67 ;)  then  proceed  a? 
directed  at  Art.  105. 


107. —  Two  right  lines  being  given,  to  find  a  third  propor- 
tional thereto.  Let  A  and  jB,  (Fig.  75,)  be  the  given  lines. 
Make  a  b  equal  to  A  ;  from  a,  draw  a  c,  at  any  angle  with  a  b  ; 
make  a  c  and  a  d  each  equal  to  B  ;  join  c  and  b  ;  from  d,  draw 
d  e,  parallel  to  c  b  ;  then  a  e  will  be  the  third  proportional  re- 
quired. That  is,  a  e  bears  the  same  proportion  to  B,  as  B  does 
to  A. 


108.—  Three  right  lines  being-  given,  to  find  a  fourth  pro 
portional  thereto.  Let  A,  Band  C,  (Fig-.  76,)  be  the  given 
lines.  Make  a  b  equal  to  A  ;  from  a,  draw  a  c,  at  any  angle 
with  a  6;  make  a  c  equal  to  B,  and  a  e  equal  to  C;  join  c  and 
6  /  from  e,  draw  ef,  parallel  to  c  b;  then  a/  will  be  the  fourth 
proportional  required.  That  is,  a  f  bears  the  same  proportion 
to  C,  as  B  does  to  A. 


AMERICAN    HOL'SE-CARPENTER.  45 

To  apply  this  problem,  suppose  the  two  axes  of  a  given  ellipsis 
and  the  longer  axis  of  a  proposed  ellipsis  are  given.  Then,  by 
this  problem,  the  length  of  the  shorter  axis  to  the  proposed  ellip- 
sis, can  be  found  ;  so  that  it  will  bear  the  same  proportion  to  the 
longer  axis,  as  the  shorter  of  the  given  ellipsis  does  to  its  longer. 
(See  also,  Art.  126.) 


109. — A  line  with  certain  divisions  being  given,  to  divide 
another,  longer  or  shorter,  given  line  in  the  same  proportion. 
Let  A,  (Fig.  77,)  be  the  line  to  be  divided, -and  B  the  line  with 
its  divisions.  Make  a  b  equal  to  B,  with  all  its  divisions,  as  at 
1,  2,  3,  &c. ;  from  a,  draw  a  c,  at  any  angle  with  a  b  ;  make  a  c 
equal  to  A  ;  join  c  and  b  ;  from  the  points,  1,  2,  3,  &c.,  draw 
lines,  parallel  to  c  b  ;  then  these  will  divide  the  line,  a  c,  in  the 
same  proportion  as  B  is  divided — as  was  required. 

This  problem  will  be  found  useful  in  proportioning  the  mem- 
bers of  a  proposed  cornice,  in  the  same  proportion  as  those  of  a 
given  cornice  of  another  size.  (See  Art.  253  and  254.)  So  of 
a  pilaster,  architrave,  &c. 


Fig.  78. 


110. — Between  two  given  right  lines..,  to  find  a  meanpjo> 
portional.  Let  A  and  B,  (Fig.  78.)  be  the  given  lines.  On 
the  line,  a  c,  make  a  b  equal  to  A,  and  b  c  equal  to  B  ;  bisect  a 
c  in  e  ;  upon  e,  with  e  a  for  radius,  describe  the  semi -circle,  a  d 


4fl  AMERICAN   HOUSE-CARPENTER. 

c  ;  at  *,  erect  I  d,  at  right  angles  toac;  then  I  d  will  be  the 
mean  proportional  between  A  and  B.  That  is,  a  5  is  to  I  du 
Id  is  to  be.  Tliis  is  usually  stated  thus— a  I  :l  br.bd' 
and  since  the  product  of  the  means  equals  the  product  of  the 
extremes,  therefore,  abxbc=Td\  This  is  shown  geometri- 
cally at  Art.  105. 

CONIC   SECTIONS. 

111.— If  a  cone,  standing  upon  a  "base  that  is  at  right  angles 
with  its  axis,  be  cut  by  a  plane,  perpendicular  to  its  base  and 
passing  through  its  axis,  the  section  will  be  an  isosceles  triangle  ; 


(e&  a  I  c,  Fig.  79 ;)  and  the  base  will  be  a  semi-circle.  If  a  cone 
be  cut  by  a  plane  in  the  direction,  e  f,  the  section  will  be  an 
ellipsis  /  if  in  the  direction,  m  I,  the  section  will  be  &  parabola  ; 
and  if  in  the  direction,  r  o,  an  hyperbola.  (See  Art.  56  to  60.)  If 
the  cutting  planes  be  at  right  angles  with  the  plane,  a  b  c,  then — 
112.— To  find  the  axes  of  the  ellipsis,  bisect  ef,  (Fig.  79,)  in 
g  ;  through  0,  draw  h  i,  parallel  to  a  b  ;  bisect  h  i  iuj;  upon 
j,  with  j  h  for  radius,  describe  the  semi-circle,  h  k  i  ;  from  ^, 
draw  g  k,  at  right  angles  to  h  i;  then  twice  g  ~k  will  be  the 
conjugate  uxis,  and  0/the  transverse. 


AMERICAN    HOUSE-CARPENTER.  47 

113. —  To  find  the  axis  and  base  of  the  parabola.  Let  m  /, 
(Fig-  79,)  parallel  to  a  c,  be  the  direction  of  the  cutting  plane 
From  m,  draw  m  d,  at  right  angles  to  a  b  ;  then  I  m  will  be  the 
axis  and  height,  and  m  d  an  ordinate  and  half  the  base ;  as  at 
Fig.  92,  93. 

114. —  To  find  the  height,  base  and  transverse  axis  of  an 
hyperbola.  Let  o  r,  (Fig.  79,)  be  the  direction  of  the  cutting 
plane.  Extend  o  r  and  a  c  till  they  meet  at  n  ;  from  o,  draw 
o  p,  at  right  angles  to  a  b;  thenro  will  be  the  height,  nrthe 
transverse  axis,  and  o  p  half  the  base  ;  as  at  Fig.  94. 


Fig.  80. 


115. —  The  axes  being  given,  to  find  the  foci,  and  to  describe 
an  ellipsis  ivith  a  string.  Let  a  b,  (Fig.  80,)  and  c  d,  be  the 
given  axes.  Upon  c,  with  a  e  or  b  e  for  radius,  describe  the  arc, 
ff;  then /and/,  the  points  at  which  the  arc  cuts  the  transverse 
axis,  will  be  the  foci.  At/  and /place  two  pins,  and  another  at  c  ; 
tie  a  string  about  the  three  pins,  so  as  to  form  the  triangle,  ffc  ; 
remove  the  pin  from  c,  and  place  a  pencil  in  its  stead ;  keeping  the 
string  taut,  move  the  pencil  in  the  direction,  eg  a;  it  will  then 
describe  the  required  ellipsis.  The  lines,  fg  and  g  /  show  the 
position  of  the  string  when  the  pencil  arrives  at  g. 

This  method,  when  performed  correctly,  is  perfectly  accurate ; 
but  the  string  is  liable  to  stretch,  and  is,  therefore,  not  so  good  to 
use  as  the  trammel.  In  making  an  ellipse  by  a  string  or  twine, 
that  kind  should  be  used  which  has  the  least  tendency  to  elasticity, 
For  this  reason,  a  cotton  cord,  such  as  chalk-lines  are  commonly 
made  of,  is  not  proper  for  the  purpose  :  a  linen,  or  flaxen  cord  is 
miuh  better. 


43  PRACTICAL    GEOMETRY. 


Fig.  81 

jig y%g  axes  being  giwn,  to  describe  an  ellipsis  with  a 

trammel.  Let  a  b  and  c  d,  (Fig.  81,)  be  the  given  axes.  Place 
the  trammel  so  that  a  line  passing  through  the  centre  of  the 
grooves,  would  coincide  with  the  axes  ;  make  the  distance  from 
the  pencil,  e,  to  the  nut,/,  equal  to  half  c  d  ;  also,  from  the  pen- 
cil, e,  to  the  nut,  g,  equal  to  half  a  b  ;  letting  the  pins  under  the 
nuts  slide  in  the  grooves,  move  the  trammel,  e  g,  in  the  direction, 
c  6  d  ;  then  the  pencil  at  e  will  describe  the  required  ellipse. 

A  trammel  maybe  constructed  thus  :  take  two  straight  strips  ot 
-  board,  and  make  a  groove  on  their  face,  in  the  centre  of  their 
width  ;  join  them  together,  in  the  middle  of  their  length,  at  right 
angles  to  one  another ;  as  is  seen  at  Fig.  81.  A  rod  is  then  to  be 
prepared,  having  two  moveable  nuts  made  of  wood,  with  a  mor- 
tice through  them  of  the  size  of  the  rod,  and  pins  under  them 
large  enough  to  fill  the  grooves.  Make  a  hole  at  one  end  of  the 
rod,  in  which  to  place  a  pencil.  In  the  absence  of  a  regular  tram- 
mel, a  temporary  one  may  be  made,  which,  for  any  short  job, 
will  answer  every  purpose.  Fasten  two  straight-edges  at  right 
angles  to  one  another.  Lay  them  so  as  to  coincide  with  the  axes 
of  the  proposed  ellipse,  having  the  angular  point  at  the  centre. 
Then,  in  a  rod  having  a  hole  for  the  pencil  at  one  end,  place  two 
brad-awls  at  the  distances  described  at  Art.  116.  While  the 
pencil  is  moved  in  the  direction  of  the  curve,  keep  the  brad-awls 
hard  against  the  straight-edges,  as  directed  for  using  the  tram- 
mel-rod, and  one-quarter  of  the  ellipse  will  be  drawn.  Then, 
by  shifting  the  straight-edges,  the  other  three  quarters  in  succes- 
sion may  lie  drawn.  If  the  required  ellipse  be  not  too  large,  a 
carpenters'-square  may  be  made  use  of,  in  place  of  the  straight- 
edges. 

An  improved  method  of  constructing  the  trammel,  is  as  fol 
lows  :  make  the  sides  of  the  grooves  bevilling  from  the  face  of 
the  stuff,  or  dove-tailing  instead  of  square.  Prepare  two  slips  ot 
wood,  each  about  two  inches  long,  which  shall  be  of  a  shape  to 
rust  fill  the  groove  when  slipped  in  at  the  end.  These,  instead  oJ 


AMERICAN    HOUSE-CARPENTER. 


pins,  are  to  be  attached  one  to  each  of  the  moveable  nuts  with 
a  screw,  loose  enough  for  the  nut  to  move  freely  about  the  screw 
as  an  axis.  The  advantage  of  this  contrivance  is,  in  preventing 
the  nuts  from  slipping  out  of  their  places,  during  the  operation 
of  describing  the  curve. 


Fig.  82. 

117. —  To  describe  an  ellipsis  by  ordinates.  Let  a  b  and  c  a, 
(Fig.  82,)  be  given  axes.  With  c  e  or  e  d  for  radius,  de- 
scribe the  quadrant, /g-  h  ;  divide  /  A,  a  e  and  e  b,  each  into  a 
like  number  of  equal  parts,  as  at  1 ,  2  and  3 ;  through  these 
points,  draw  ordinates,  parallel  to  c  d  and/^  ;  take  the  distance, 
1 1,  and  place  it  at  1  /,  transfer  2  ;  to  2  m,  and  3  A:  to  3  n  ;  through 
the  points,  a,  n,  m,  I  and  c,  trace  a  curve,  and  the  ellipsis  will 
be  completed. 

The  greater  the  number  of  divisions  on  a  e,  &c.,  in  this- and 
the  following  problem,  the  more  points  in  the  curve  can  be  found, 
and  the  more  accurate  the  curve  can  be  traced.  If  pins  are 
placed  in  the  points,  n,  m,  /,  &c.,  and  a  thin  slip  of  wood  bent 
around  by  them,  the  curve  can  be  made  quite  correct.  This 
method  is  mostly  used  in  tracing  face-moulds  for  stair  hand- 
railing. 


118. —  To  describe  an  ellipsis  by  intersection  of  lines. 

7 


Lei 


Ri) 


PRACTICAL    GEOMETRY. 


a  b  and  c  d,  (Fig.  83,)  be  given  axes.  Through  c  draw  /  «-, 
parallel  tc  a  b  ;  from  a  and  6,  draw  a  /and  b  g,  at  right  anglns 
to  a  6  ;  divide  f  a,  g  b,  a  e  and  e  6,  each  into  a  like  number  of 
equal  parts,  as  at  1,  2,  3  and  o,  o,  o  ;  from  1,  2  and  3,.  draw  lines 
to  c  ;  through  o,  o  and  o,  draw  lines  from  d,  intersecting  those 
drawn  to  c  ;  then  a  curve,  traced  through  the  points,  t,  i,  t,  will 
be  that  of  an  ellipsis. 


Where  neither  trammel  nor  string  is  at  hand,  this,  perhaps,  Is 
the  mcst  ready  method  of  drawing  an  ellipsis.  The  divisions 
should  be  small,  where  accuracy  is  desirable.  By  this  method, 
an  ellipsis  may  be  traced  without  the  axes,  provided  that  a  diame- 
ter and  its  conjugate  be  given.  Thus,  a  b  and  c  d,  (Fig-  84,)  are 
conjugate  diameters :  /  g  is  drawn  parallel  to  a  6,  instead  of 
being  at  right  angles  to  c  d  ;  also,  /  a  and  g  b  are  drawn  parallel 
to  c  d,  instead  of  being  at  right  angles  to  a  b. 


119.—  To  describe  an  ellipsis  by  intersecting-  arcs.     Let  a  b 


AMERICAN    HOUSE-CARPENTER. 


51 


ana  c  d,  (Fig:  85,)  be  given  axes.  Between  one  of  the  foci,/ 
ai.d/  and  the  centre,  e,  mark  any  number  of  points,  at  random, 
as  1,  2  and  3  ;  up-ia/and/,  with  b  I  for  radius,  describe  arcs  at 
£T?  £•>  £  and  S"  i  uPon/  and/,  with  a  1  for  radius,  describe  arcs  inter- 
secting the  others  at  g,g,g  and  g;  then  these  points  of  intersection 
will  be  in  the  curve  of  the  ellipsis.  The  other  points,  h  and  i,  are 
found  in  like  manner,  viz :  h  is  found  by  taking  b  2  for  one  radius, 
and  a  2  for  the  other ;  i  is  found  by  taking  b  3  for  one  radius,  and 
a  3  for  the  other,  always  using  the  foci  for  centres.  Then  by 
tracing  a  curve  through  the  points,  c,  g,  h,  t,  6,  &c.,  the  ellipse 
will  be  completed. 

This  problem  is  founded  upon  the  same  principle  as  that  of  the 
string.  This  is  obvious,  when  we  reflect  that  the  length  of  the 
string  is  equal  to  the  transverse  axis,  added  to  the  distance  between 
the  foci.  See  Fig.  80 ;  in  which  c  /  equals  a  e,  the  half  of  the 
transverse  axis. 


Fig.  86. 


120. —  To  describe  a  figure  nearly  in  the  shape  of  an  ellip- 
sis, by  a  pair  of  compasses.  Let  a  b  and  c  d,  (Fig.  86,)  be 
given  axes.  From  e,  draw  c  e,  parallel  to  a  b  ;  from  a,  draw  a  e, 
parallel  to  c  d;  join  e  and  d;  bisect  e  a  in/;  join/ and  c,  inter- 
secting e  d  in  i;  bisect  i  c  in  o  ;  from  o,  draw  og,  at  right  angles 
to  i  c,  meeting  c  d  extended  to  g  ;  join  i  and  g,  cutting  the  trans- 
verse axis  in  r  ;  make  h  j  equal  to  h  g,  and  h  k  equal  to  h  r  , 
from  J,  through  r  and  &,  drawj  m  and  j  n  ;  also,  from  g,  through 
k,  draw  g  I;  upon  g  and  /,  with  g  c  for  radius,  describe  the 


52  PRACTICAL    GEOMETRY. 

arcs,  i  I  and  m  n ;  upon  r  an  1  k,  with  r  a  for  radius,  describe 
the  arcs,  m  i  and  /  n  ,  this  will  complete  the  figure. 

When  the  axes  are  proportioned  to  one  another  as  2  to  3,  the 
extremities,  c  and  d,  of  the  shortest  axis,  will  be  the  centres  for 
describing  the  arcs,  i  /and  m  n;  and  the  intersection  of  e  d  with 
the  transverse  axis,  will  be  the  centre  for  describing  the  arc,  m  i, 
&c.  As  the  elliptic  curve  is  continually  changing  its  course  from 
that  of  a  circle,  a  true  ellipsis  cannot  be  described  with  a  pair  ot 
compasses.  The  above,  therefore,  is  only  an  approximation. 


Fig.  87. 


121. —  To  draw  an  oval  in  the  proportion,  seven  by  nine. 
Let  c  rf,  (Fig.  87,)  be  the  given  conjugate  axis.  Bisect  c  d  in  o, 
and  through  o,  draw  a  6,  at  right  angles  to  c  d  ;  bisect  c  o  in  e  , 
upon  o,  with  o  e  for  radius,  describe  the  circle,  e  f  g  h  ;  from  e, 
through  h  and/,  draw  e  j  and  e  i  ;  also,  from  §•,  through  h  and/, 
draw  g  k  and  g  I ;  upon  g,  with  g  c  for  radius,  describe  the  arc, 
k  I ;  upon  e,  with  e  d  for  radius,  describe  the  arc,  j  i  ;  upon  h  and 
/",  with  h  k  for  radius,  describe  the  arcs,  j  k  and  I  i  ;  this  will 
complete  the  figure. 

This  is  an  approximation  to  an  ellipsis  ;  and  perhaps  no 
method  can  be  found,  by  which  a  well-shaped  oval  can  be  drawn 
with  greater  facility.  By  a  little  variation  in  the  piocess,  ovals 
of  different  proportions  may  be  obtained.  If  quarter  cf  the  trans- 
verse axis  is  taken  for  the  radius  of  the  circle,  efg  h,  one  will  be 
drawn  in  the  proportion,  five  by  seven. 


AMERICAN    HOUSE-CARPENTER. 


53 


Fig.  88. 


122. —  To  draw  a  tangent  to  an  ellipsis.  Let  abed,  (Fig. 
38,)  be  the  given  ellipsis,  and  d  the  point  of  contact.  Find  the 
foci,  (Art.  115,)/and/,  and  from  them,  through  d,  draw/e  and 
f  d;  bisect  the  angle,  (Art.  77,)  e  d  o,  with  the  line,  sr;  then 
s  r  will  be  the  tangent  required. 


Fig   89 


123. — An  ellipsis  with  a  tangent  given,  to  detect  the  point 
of  contact.  L,etagbf,  (Fig.  89,)  be  the  given  ellipsis  and  tan- 
gent. Through  the  centre.,  e,  draw  a  b,  parallel  to  the  tangent ; 
any  where  between  e  and/,  draw  c  d,  parallel  to  a  b  ;  bisect  c  d  in 
o  ;  through  o  and  e,  draw  f  g  ;  then  g  will  be  the  point  of  con- 
tact required. 

124. — A  diameter  of  an  ellipsis  given,  to  find  its  conjugate. 
Let  a  b,  (Fig.  89,)  be  the  given  diameter.  Find  the  line,fg,  by 
Ihe  last  problem;  thenfg  will  be  th  3  diameter  required. 


fig.  90. 

125. — Any  diameter  and  its  conjugate  being  given,  tc  as- 
certain the  two  axes,  and  thence  to  describe  the  ellipsis.  i_<et 
a  b  and  c  d,  (Fig.  90,)  be  the  given  diameters,  conjugate  to  one 
another.  Through  c,  draw  ef,  parallel  to  a  b  ;  from  c,  draw  c 
g,  at  right  angles  to  ef;  make  c  g  equal  to  a  A  or  A  b  /  join  g 
and  A  ;  upon  g,  with  £•  c  for  radius,  describe  the  arc,  i  k  c  j  ; 
upon  A,  with  the  same  radius,  describe  the  arc,  I  n  ;  through  the 
int^rsedtions,  I  and  n,  draw  n  o,  cutting  the  tangent,  e/  in  o  ; 
upon  o,  with  o  gfor  radius,  describe  the  semi-circle,  e  ig  f ;  join 
e  and  if,  also  g  and/  cutting  the  arc,  i  c  j,  in  k  and  t ;  from  e, 
through  A,  draw  e  m,  also  from/  through  A,  draw//?/  from  k 
and  £,  draw  A:  r  and  £  5,  parallel  to  g  A,  cutting  e  ni  in  r,  and//> 
in  a  ;  make  A  w  equal  to  A  r,  and  A  p  equal  to  As/  then  r  in 
and  s  p  will  be  the  axes  required,  by,  which  the  ellipsis  may  be 
drawn  in  the  usual  way. 

126. —  To  describe  an  ellipsis,  whose  axes  shall  be  propor- 
tionate to  the  axes  of  a  larger  <K  smaller  given  one.  Let  a 
cbd,  (Fig.  91,)  be  the  given  ellipsis  and  axes,  and  ij  the  trans- 
verse axis  of  a  proposed  smaller  one.  Join  a  and  c  /  trom  i 
Iraw  t.e,  parallel  to  a  c  /  make  o  f  equal  to  o  e  /  then  ef  will  be 


AMERICAN    HOUSE-CARPENTER. 


55 


Fig.  91. 


the  conjugate  axis  required,  and  will  bear  the  same  proportion  to 
•  j,ascd  does  to  a  b.     (See  Art.  108.) 


123       132 


0 

/ 


\ 


\ 


d      1     2     3     m     3     2     1        d 

Fig.  92. 

127. —  To  describe  a  parabola  by  intersection  of  lines.  Lei 
m  I,  (Fig:  92,)  be  the  axis  and  height,  (see  Fig.  79,)  and  d  d}  a 
double  ordinate  and  base  of  the  proposed  parabola.  Through  L 
draw  a  a,  parallel  to  d  d  ;  through  d  and  d,  draw  d  a  and  d  a, 
parallel  to  rn  I ;  divide  a  d  and  d  m,  each  into  a  like  number  oi 
equal  parts  ;  from  each  point  of  division  in  d  m,  draw  the  lines, 
1  1,  2  2,  &c.,  parallel  to  ml;  from  each  point  of  division  in  d 
a,  draw  lines  to  I ;  then  a  curve  traced  through  the  points  ot 
intersection,  o,  o  and  o,  will  be  that  of  a  parabola. 

127,  a.— Another  method.  Let  m  /,  (Fig.  93,)  be  the  axis  and 
height,  and  d  d  the  base.  Extend  m  Z,  and  make  I  a  equal  to  in 
i ;  join  a  and  c?,  and  a  and  d ;  divide  a  d  and  a  d,  each  into  a 
liice  number  of  equal  parts,  as  at  1,  2,  3,  &c. ;  join  1  and  1,  2  and 
2,  &c.,  and  the  parabola  will  be  completed 


56 


PRACTICAL    GEOMETRY. 


Fig.  93. 


123. —  To  describe  an  hyperbola  by  intersection  of  lines. 
Let  r  o,  (Fig.  94,)  be  the  height,  p  p  the  base,  and  n  r  the  trans- 
verse axis.  (See  Fig.  79.)  Through  r,  draw  a  a,  parallel  to  p 
p  ;  irom  p,  draw  a  p,  parallel  to  r  o  ;  divide  a  p  and  p  o,  each 
into  a  like  number  of  equal  parts  ;  from  each  of  the  points  of  di- 
visions in  the  base,  draw  lines  to  n  ;  from  each  of  the  points  of 
division  in  a  p,  draw  lines  to  r  ;  then  a  curve  traced  through  the 
points  of  intersection,  o,  o,  &c.,  will  be  that  of  an  hyperbola. 

The  parabola  and  hyperbola  afford  handsome  curves  for  various 
moildings. 


DEMONSTRATIONS. 


129. — To  impress  more  deeply  upon  the  mind  of  the  learnei 
some  of  the  more  important  of  the  preceding  problems,  and  to 
indulge  a  very  common  and  praiseworthy  curiosity  to  discover 
the  cause  of  things, 'are  some  of  the  reasons  why  the  following 
exercises  are  introduced.  In  all  reasoning,  definitions  are  ne- 
cessary ;  in  order  to  insure,  in  the  minds  of  the  proponent  and 
respondent^  identity  of  ideas.  A  corollary  is  an  inference  deduced 
from  a  previous  course  of  reasoning.  An  axiom  is  a  proposition 
evident  at  first  sight.  In  the  following  demonstrations,  there  are 
many  axioms  taken  for  granted ;  (such  as,  things  equal  to  the 
same  thing  are  equal  to  one  another,  &c. ;)  these  it  was  thought 
not  necessary  to  introduce  in  form. 


Fig.  95. 


130. — Definition.     If  a  straight  line,  as  a  b,  (Fig.  95,)  stand 
upon  another  straight  line,  as  c  d.  so  that  the  two  angles  made  at 
8 


PRACTICAL    GEOMETRY. 


the  point,  b,  are  equal—  a  b  c  to  a  b  d,  (see  note  to  Art.  27,)  then 
each  of  the  two  angles  is  called  a  right  angle. 

131.  _  Definition.  The  circumference  of  every  circle  is  sup- 
posed to  be  divided-  into  360  equal  parts,  called  degrees  ;  hence 
a  serai-circle  contains  180  degrees,  a  quadrant  90,  &c. 


Fig.  981 


132. — Definition.  The  measure  of  an  angle  is  the  number  of 
degrees  contained  between  its  two  sides,  using  the  angular  point 
as  a  centre  upon  which  to  describe  the  arc.  Thus  the  arc,  c  e, 
(Fig.  96,)  is  the  measure  of  the  angle,  c  b  e  ;  e  a,  of  the  angle, 
e  b  a  ;  and  a  c?,  of  the  angle,  a  b  d. 

133. —  Corollary.  As  the  two  angles  at  6,  (Fig.  95,)  are  right 
angles,  and  as  the  semi-circle,  c  a  c?,  contains  180  degrees,  (Art. 
131,)  the  measure  of  two  right  angles,  therefore,  is  180  degrees  ; 
of  one  right  angle,  90  degrees  ;  of  half  a  right  angle,  45 ;  of 
one-third  of  a  right  angle,  30,  &c. 

134. — Definition.  In  measuring  an  angle,  (Art.  132,)  no  re- 
gard is  to  be  kad  to  the  length  of  its  sides,  but  only  to  the  degree 
of  their  inclination.  Hence  equal  angles  are  such  as  have  the 
same  degree  of  inclination,  without  regard  to  the  length  of  their 
sides. 


7 
135. — Axiom.    If  two  straight  lines,  parallel  to  one  another, 


AMERICAN    HOUSE-CARPENTER.  59 

as  a  b  and  c  d,  (Fig.  97,)  stand  upon  another  straight  line,  as  e/( 
the  angles,  a  b  f  and  c  d  /,  are  equal ;  and  the  angle,  a  b  e,  is 
equal  to  the  angle,  c  d  e. 

136. — Definition.  If  a  straight  line,  as  a  b,  (Fig.  96,)  stanc! 
obliquely  upon  another  straight  line,  as  cd,  then  one  of  the  an- 
gles, as  a  6  c,  is  called  an  obtuse  angle,  and  the  other,  as  a  b  d, 
an  acute  angle. 

137. — Axiom.  The  two  angles,  a  b  d  and  a  b  c,  (Fig.  96,)  are 
together  equal  to  two  right  angles,  (Art.  130,  1 33  ;)  also,  the 
three  angles,  a  b  d,  e  b  a  and  cb  e,  are  together  equal  to  two  right 
angles. 

138. — Corollary.  Hence  all  the  angles  that  can  be  made  upon 
one  side  of  a  line,  meeting  in  a  point  in  that  line,  are  together 
equal  to  two  right  angles. 

139. — Corollary.  Hence  all  the  angles  that  can  be  made  on 
both  sides  of  a  line,  at  a  point  in  that  line,  or  all  the  angles  that 
can  be  made  about  a  point,  are  together  equal  to  four  right  angles. 


140. — Proposition.  If  to  each  of  two  equal  angles  a  third 
angle  be  added,  their  sums  will  be  equal.  Let  a  b  cand  d  e/, 
(Fig.  98,)  be  equal  angles,  and  the  angle,  i  j  k,  the  one  to  be 
added.  Make  the  angles,  g  b  a  and  hed,  each  equal  to  the  given 
angle,  ij  k  ;  then  the  angle,  g  b  c,  will  be  equal  to  the  angle,  h  e 
f;  for,  if  a  b  c  and  d  e/be  angles  of  90  degrees,  and  i  j  k,  30, 
then  the  angles,  g  b  c  and  h  ef,  will  te  each  equal  to  90  and 
30  added,  viz :  120  degrees. 


PRACTICAL    GEOMETRY. 
*. 


141. — Proposition.  Triangles  that  have  two  of  their  sides 
and  the  angle  contained  between  them  respectively  equal,  have 
also  their  third  sides  and  the  two  remaining  angles  equal ;  arid 
consequently  one  triangle  will  every  way  equal  the  other.  Let  a 
b  c,  (Fig.  99,)  and  d  e/be  two  given  triangles,  having  the  angle 
at  a  equal  to  the  angle  at  e?,  the  side,  a  b,  equal  to  the  side,  d  e, 
and  the  side,  a  c,  equal  to  the  side,  df;  then  the  third  side  of 
one,  b  c,  is  equal  to  the  third  side  of  the  other,  e  f;  the  angle  at  b 
is  equal  to  the  angle  at  e,  and  the  angle  at  c  is  equal  to  the  angle 
at/.  For,  if  one  triangle  be  applied  to  the  other,  the  three  points, 
6,  a,  c,  coinciding  with  the  three  points,  e,  </,  /,  the  line,  b  c,  must 
coincide  with  the  line,  e  f;  the  angle  at  b  with  the  angle  at  e  ; 
the  angle  at  c  with  the  angle  at/;  and  the  triangle,  b  a  c,  be  every 
way  equal  to  the  triangle,  e  df. 


142. — Proposition.  The  two  angles  at  the  base  of  an  isoceles 
triangle  are  equal.  Let  a  b  c,  (Fig.  100,)  be  an  isoceles  triangle, 
of  which  the  sides,  a  b  and  a  c,  are  equal.  Bisect  the  angle,  (Art. 


AMERICAN    HOUSE-CARPENTER.  (JJ 

77,)  b  a  c,  by  the  line,  a  d.  Then  the  line,  b  a,  being  equal  to 
the  line,  a  c  ;  the  line,  a  d,  of  the  triangle,  A,  being  equal  to  the 
line,  a  d,  of  the  triangle,  B,  being  common  to  each  ;  the  angle,  b 
a  d,  being  equal  to  the  angle,  d  a  c  ;  the  line,  b  d,  must,  accord- 
ing to  Art.  141,  be  equal  to  the  line,  d  c  ;  and  the  angle  at  b  urns* 
be  equal  to  the  angle  at  c. 


143. — Proposition.  A  diagonal  crossing  a  parallelogram  di- 
vides it  into  two  equal  triangles.  Let  abed,  (Fig.  101,)  be  a 
given  parallelogram,  and  b  c,  a  line  crossing  it  diagonally.  Then, 
as  a  c  is  equal  to  b  d,  and  a  b  to  c  d,  the  angle  at  a  to  the  angle 
at  d,  the  triangle,  A,  must,  according  to  Art.  141,  be  equal  to  the 
triangle,  B. 


Fig.   102. 

144. — Proposition.  Let  abed,  (Fig.  102,)  be  a  given  pa« 
rallelogram,  and  b  c  a  diagonal.  At  any  distance  between  a  b  and 
c  d,  draw  e  /,  parallel  to  a  b  ;  through  the  point,  g,  the  intersection 
of  the  lines,  b  c  and  e/,  draw  h  t,  parallel  to  b  d.  In  every  paral- 
lelogram thus  divided,  the  parallelogram,  A,  is  equal  to  the  paral- 
lelogram, B.  According  to  Art.  143,  the  triangle,  a  b  c,  ia 
equal  to  the  triangle,  bed;  the  triangle,  C,  to  the  triangle,  D  ; 
and  E  to  F;  this  being  the  case,  take  D  and  F  from  the  triangle, 
b  c  d,  and  C  and  E  from  the  triangle,  a  b  c,  and  what  remains 


G-2 


PRACTICAL    GEOMETRY. 


in  one  must  be  tqual  to  what  remains  in  the  other;  therefore,  thf. 
parallelogram,  A,  is  equal  to  the  parallelogram,  B. 


Fig.  103. 


145. — Proposition.  Parallelograms  standing  upon  the  same 
base  and  between  the  same  parallels,  are  equal.  Let  abed  and 
efcd,  (Fig.  103,)  be  given  parallelograms,  standing  upon  the 
same  base,  c  d,  and  between  the  same  parallels,  a  f  and  c  d. 
Then,  a b  and  ef  being  equal  to  c  d,  are  equal  to  one  another; 
6  e  being  added  to  both  a  b  and  ef,  a  e  equals  bf;  the  line,  ac, 
being  equal  to  b  d,  and  a  e  to  b  f,  and  the  angle,  t  a  e,  being 
equal,  (Art.  135,)  to  the  angle,  d  bf,  the  triangle,  a  e  c,  must  be 
equal,  (Art.  141,)  to  the  triangle,  bfd;  these  two  triangles  being 
equal,  take  the  same  amount,  the  triangle,  beg,  from  each,  and 
what  remains  in  one,  a  b  g  c,  must  be  equal  to  what  remains  in 
the  other,  efdg;  these  two  quadrangles  being  equal,  add  the 
same  amount,  the  triangle,  c  g  d,  to  each,  and  they  must  still  be 
equal ;  therefore,  the  parallelogram,  a  b  c  d,  is  equal  to  the  paral- 
lelogram, efcd. 

146. — Corollary.  Hence,  if  a  parallelogram  and  triangle  stand 
upon  the  same  base  and  between  the  same  parallels,  the  parallelo- 
gram will  be  equal  to  double  the  triangle.  Thus,  the  paral- 
lelogram, a  d,  (Fig.  103,)  is  double,  (Art.  143,)  the  triangle, 
ced. 

147. — Proposition.  Let  abed,  (Fig.  104,)  be  a  given  quad- 
rangle with  the  diagonal,  a  d.  From  b,  draw  b  e,  parallel  to  a  d, 
extend  cdtoe  ;  join  a  and  e  ;  then  the  triangle,  a  ec,  will  be  equal 
in  area  to  the  quadrangle,  abed.  Since  the  triangles,  a  d  b  and 
ad  e,  stand  uj  on  the  same  base,  a  d,  and  bet  ween  the  same  paral- 


AMERICAN    HOUSE-CARPENTER. 


63 


Fig.  104. 


lels,  a  d  and  b  e,  they  are  therefore  equal,  (Art.  145,  146  ;}  and 
since  the  triangle,  C,  is  common  to  both,  the  remaining  triangles,  A 
and  B,  are  therefore  equal ;  then  B  being  equal  to  A,  the  triangle, 
a  e  c,  is  equal  to  the  quadrangle,  abed. 


Fig.  105. 

148. — Proposition.  If  two  straight  lines  cut  each  other,  as 
a  b  and  c  d,  (Fig.  105,)  the  vertical,  or  opposite  angles,  A  and 
C,  are  equal.  Thus,  a  e,  standing  upon  c  d,  forms  the  angles, 
B  and  C,  which  together  amount,  (Art.  137,)  to  two  right  angles  ; 
in  the  same  manner,  the  angles,  A  and  J5,  form  two  right  angles  ; 
since  the  angles,  A  and  £,  are  equal  to  B  and  C,  take  the  same 
amount,  the  angle,  J5,  from  each  pair,  and  what  remains  of  one 
pair  is  equal  to  what  remains  of  the  other ;  therefore,  the  an- 
gle, A,  is  equal  to  the  angle,  C.  The  same  can  be  proved  ot 
the  opposite  angles,  B  and  D. 

149. — Proposition.  The  three  angles  of  any  triangle  are 
equal  to  two  right  angles.  Let  a  b  c,  (Fig.  106,)  be  a  given  tri- 
angle, with  its  sides  extended  to/,  e}  and  c/,  and  the  line,  eg. 


64  PRACTICAL    GEOMETRY. 


fig.  106. 

drawn  parallel  to  b  e.  As  g  c  is  parallel  to  e  6,  the  angle,  g  c  dt 
is,  equal,  (Art.  135,)  to  the  angle,  e  b  d ;  as  the  lines,  f  c  and  b  e, 
cut  one  another  at  a,  the  opposite  angles,  /  a  e  and  b  a  c,  are 
equal,  (Art.  148  :)  as  the  angle,  /  a  e,  is  equal,  (Art.  135,)  to  the 
angle,  a  eg,  the  angle,  a  c  g,  is  equal  to  the  angle,  b  a  c  ;  there- 
fore, the  three  angles  meeting  at  c,  are  equal  to  the  three  angles 
of  the  triangle,  a  b  c  ;  and  since  the  three  angles  at  c  are  equal, 
(Art.  137,)  to  two  right  angles,  the  three  angles  of  the  triangle,  a 
b  c,  must  likewise  be  equal  to  two  right  angles.  Any  triangle 
can  be  subjected  to  the  same  proof. 

150. — Corollary.  Hence,  if  one  angle  of  a  triangle  be  a  right 
angle,  the  other  two  angles  amount  to  just  one  right  angle. 

151. — Corollary.  If  one  angle  of  a  triangle  be  a  right  angle, 
and  the  two  remaining  angles  are  equal  to  one  another,  these  are 
each  equal  to  half  a  right  angle. 

152. — Corollary.  If  any  two  angles  of  a  triangle  amount  to 
a  right  angle,  the  remaining  angle  is  a  right  angle. 

153. — Corollary.  If  any  two  angles  of  a  triangle  are  togethei 
equal  to  the  remaining  angle,  that  remaining  angle  is  a  right 
angle. 

154. — Corollary.  If  any  two  angles  of  a  triangle  are  each 
equal  to  two-thirds  of  a  right  angle,  the  remaining  angle  is  alsc 
equal  to  two-thirds  of  a  right  angle. 

155. —  Corollary.  Hence,  the  angles  of  an  equi-lateral  trian- 
gle, are  each  equal  to  two-thirds  of  a  right  angle. 


AMERICAN   HOUSE-CJRPKNTEB. 


r  ,-   ;  :. 


156. — Proposition.  If  from  tLe  extremities  of  the  diameter  of 
a  semi-circle,  two  straight  lines  be  drawn  to  any  point  in  tne  cir- 
cumference, the  angle  formed  by  them  af  that  point  wui  be  a 
rignt  angle.  Let  a  b  e,  (Ffff.  107,)  be  a  given  semi-circle ,  and 
mb  and  6  e,  lines  drawn  from  the  extremities  of  the  diameter,  a 
c,  to  the  given  point,  b  ;  the  angle  formed  at  mat  point  by  these 
lines,  is  a  right  angle.  Join  the  point,  6,  and  the  centre,  d  :  the 
lines,  d  a,  d  b  and  d  c,  being  radii  of  the  same  circle,  are  equal ; 
the  angle  at  a  is  therefore  equal,  (Art.  142,)  to  the  angle,  a  b  d, 
also,  the  angle  at  c  is,  for  the  same  reason,  equal  to  the  angte,  d  I 
e:  the  angle,  a  6  e,  being  equal  to  the  angles  at  a  and  c  taken 
together,  must  therefore,  (Art.  153,)  be  a  right  angle. 


,57. — Proposition,  The  square  of  the  hypothenu&e  of  a 
right-angled  triangle,  is  equal  to  the  squares  of  the  two  remaining 
•sides.  Let  a  ft  e,  (Fig.  108.)  be  a  given  right-angled  triangle, 
having  a  square  formed  on  each  of  its  sides:  then,  the  square,  b  e,  is 
equal  to  the  squares,  h e  and  g  6,  taken  together.  Thiscanbe 


66  PRACTICAL     GEOMETRY. 

proved  by  showing  that  the  parallelogram,  b  I,  is  equal  to  the  square, 
n-  b  ;  and  that  the  parallelogram,  c  /,  is  equal  to  the  square,  h  c.  The 
angle,  c  b  d,  is  a  right  angle,  and  the  angle;  a  &/,  is  a  right  angle  ; 
add  to  each  of  these  the  angle,  a  b  c  ;  then  the  angle,/  b  c,  will  evi- 
dently be  equal,  (Art.  140,)  to  the  angle,  a  b  d;  the  triangle,/6  c. 
and  the  square,  g  b,  being  both  upon  the  same  base,  fb,  and  between 
the  same  parallels,  /  b  andg-  c,  the  square,  g  b,  is  equal,  (Art.  146,) 
to  twice  the  triangle./ b  c;  the  triangle,  a  b  d,  and  the  parallelo- 
gram, b  I,  being  both*  upon  the  same  base,  b  d,  and  between  the 
same  parallels,  b  d  and  a  I,  the  parallelogram,  6  I,  is  equal  to  twice 
the  triangle,  a  b  d;  the  triangles,/ b  c  and  a  b  d,  being  equal  to 
one  another,  (Art.  141,)  the  square,  g  b,  is  equal  to  the  parallelo- 
gram, b  Z,  either  being  equal  to  twice  the  triangle,  fb  c  or  a  b  d. 
The  method  of  proving  h  c  equal  to  c  /  is  exactly  similar — thus 
proving  the  square,  b  e,  equal  to  the  squares,  h  c  and  g  b,  taken 
together. 

This  problem,  which  is  the  47th  of  the  First  Book  of  Euclid 
is  said  to  have  been  demonstrated  first  b^  Pythagoras.  It  is  sta 
led,  (but  the  story  is  of  doubtful  authority,)  that  as  a  thank-offer 
ing  for  its  discovery  he  sacrificed  a  hundred  oxen  to  the  gods. 
From  this  circumstance,  it  is  sometimes  called  the  hecatomb  pro- 
blem. It  is  of  great  value  in  the  exact  sciences,  more  especially 
in  Mensuration  and  Astionomy,  in  which  many  otherwise  intri- 
cate calculations  are  by  it  made  easy  of  solution. 

158. — Proposition.  In  a  segment  of  a  'circle,  the  versed  sine 
equals  the  radius,  less  the  square  root  of  the  difference  of  the 
squares  of  the  radius  and  half-chord.  That  is,  the  versed  sine, 
a  c,  (Fig.  109,)  equals  a  J,  less  c  b.  Now  a  b  is  radius,  hence 
the  radius,  minus  c  5,  equals  a  c,  the  versed  sine.  To  find  the 
value  of  G  &,  it  will  be  observed  that  c  b  is  the  side  of  tlie  square, 
cf,  while  the  radius  b  d  is  the  side  of  the  square,  b  fi,  and  the 
half-chord,  c  d,  is  the  side  of  the  square,  c  e ;  also,  that  these 
three  squares  are  made  upon  the  three  sides  of  the  riglit  angled 


AMERICAN   HOUSE-CARPENTER. 


Fig  109. 

triangle,  I  c  <?,  and  the  square,  I  A,  is  therefore  equal  to  the  two 
squares,  o  e  and  c  /",  (Art.  157 ;)  therefore,  the  square,  c  f,  is 
equal  to  the  square,  I  A,  minus  the,  square,  c  e  ' — or,  is  equal  to 
the  difference  of  the  squares  on  I  d  and  c  d.  Consequently  the 
square  root  of  cf  is  e<|ual  to  the  square  root  of  the  difference 
of  the  squares  on  5  d  and  c  d  j  and  since  c  5  is  the  square  root 
of  c/,  therefore  c  ~b  equals  the  square  root  of  the  difference  of 
the  squares  on  I  d  and  c  d — or,  equals  the  square  root  of  the 
difference  of  the  squares  of  the  radius  and  the  half-chord. 
Having  found  an  expression  for  the  value  of  c  5,  it  remains 
merely  to  deduct  this  value  from  the  radius,  and  the  residue 
equals  the  versed  sine  ;  for,  as  before  stated,  the  versed  sine,  a  c, 
equals  the  radius,  a  5,  minus  c  1>  /  therefore,  the  versed  sine 
equals  the  radius,  minus  the  square  root  of  the  difference  of 
the  squares  on  the  radius  and  half-chord.  The  rule  expressed 
algebraically  is  v=r—Vr*—*dii  where  v  is  the  versed  sine,  r  the 
radius,  and  -a  the  half-chord.  It  is  read,  v  equals  r,  minus  the 
square  root  of  the  difference  of  the  squares  of  r  and  a. 

159. — Proposition.  In  an  equilateral  octagon  the  semi- 
diagonal  of  a  circumscribed  square,  having  its  sides  coincident 
with  four  of  the  s?des  of  the  octagon,  equals  the  distance  along 


PRACTICAL   GEOMETRY 


Fig.  110. 

a  side  of  the  square  from  its  corner  to  the  more  remote  angle 
of  the  octagon  occurring  on  that  side  of  the  square.  To  prove 
this,  it  need  only  to  be  shown  that  the  triangle,  a  o  d,  (Fig. 
110,)  is  an  isosceles  triangle  having  its  sides  a  o  and  a  d,  equal. 
The  octagon  being  eqni-lateral,  it  is  also  equi-angular,  therefore 
the  angles,  b  c  o,  e  c  o,  a  d  o,  &c.,  are  all  equal.  Of  the  right- 
angled  triangle,  fe  c,fc  andfe  being  equal,  the  two  angles,  fe  c 
and  y  c  e  are  equal,  (Art.  142,)  and  are  therefore,  (Art.  151,) 
each  equal  to  half  a  right  angle.  In  like  manner  it  may  be 
shown  that  /  a  b  and/  b  a  are  also  each  equal  to  half  a  right 
angle.  And  since  f  e  c  and  f  a  b  are  equal  angles,  therefore 
the  lines  e  c  and  a  b  are  parallel,  (Art.  135,)  and  hence  the 
angles,  e  c  o  and  a  o  d,  are  equal.  These  being  equal,  and  the 
angles  e  c  o  and  ado  being,  by  construction,  equal,  as  before 
shown,  therefore  the  angles  a  o  d  and  ado  are  equal,  and  con- 
sequently the  lines  a  o  and  a  d  are  equal.  (Art.  142.) 

160. — Proposition.  An  angle  at  the  circumference  of  a 
circle  is  measured  by  half  the  arc  that  subtends  it :  that  is, 
the  angle  a  be,  (Fig.  Ill,)  is  equal  to  half  the  angle  a  d  c. 
Through  the  centre,  d>  draw  the  diameter,  b  e.  The  triangle 
a  b  d  is  an  isosceles  triangle,  a  d  and  b  d  being  radii,  and  there- 
fore equal ;  hence  the  two  angles,  dab  and  d  b  a,  are  equal. 


AMERICAN   HOrSE-CARPENTEB. 


Fig.  111. 

(Art.  142,)  and  the  sum  of  these  two  angles  is  equal  to  the 
angle  a  d  e,  (Art.  149,)  and  therefore  one  of  them,  a  5  d,  is 
equal  to  the  half  of  a  d  e.  The  angles  a  d  e  and  a  b  d  (or 
ale}  are  both  subtended  by  the  arc  a  e.  Now,  since  the  angle, 
a  d  6,  is  measured  by  the  arc  a  6,  which  subtends  it,  therefore 
the  half  of  the  angle,  a  d  0,  would  be  measured  by  the  half 
of  the  arc  a  e  /  and  since  a  J)  d  is  equal  to  the  half  of  a  d  0, 
therefore  a  5  d,  or  a  I  e,  is  measured  by  the  half  of  the  arc  a  e. 
It  may  be  shown  in  like  manner  that  the  angle  e  I  c  is  mea- 
sured by  half  the  arc  e  c,  and  hence  it  follows  that  the  angle, 
a  1)  c,  is  measured  by  half  the  arc,  a  c,  that  subtends  it. 

161. — Proposition.  In  a  circle,  all  the  inscribed  angles, 
a  J>  c,  (Fig.  112,)  which  stand  upon  the  same  side  of  the  chord 
d  e,  are  equal.  For  each  angle  is  measured  by  half  the  arc 
df  e,  (Art.  160,)  hence  the  angles  are  all  equal. 

162. — Corollary.  Equal  chords,  in  the  same  circle,  subtend 
equal  angles. 

163. — Proposition.  The  angle  formed  by  a  chord  and  tan- 
gent is  equal  to  any  inscribed  angle  in  the  opposite  segment 


7C 


PRACTICAL   GEOMETET. 


Fig.  112. 


Fig.  118. 

of  the  circle ;  that  is,  the  angle  J9,  (Fig.  113,)  equals  tlie  anglb 
A.  Let  cf  be  the  chord,  and  a  5  the  tangent ;  draw  the  dia- 
meter, d  G  j  then  d  c  5  is  a  right  angle,  also  d  f  c  is  a  right 
angle.  (Art.  156.)  The  angles  A  and  B  together  equal  a 
right  angle,  (Art.  150 ;)  also  the  angles  B  and  D  together 
equal  a  right  angle,  (equal  the  angle  d  c  I ;)  therefore  the  sum 
of  A  and  JB  equals  the  sum  of  B  and  D.  From  each  of  these 
two  equals,  making  the  like  quantity  J?,  the  remainders,  A  and 


AMERICAN    HOUSE-CARrEXTEK. 


•ri 


2),  are  equal.  Thus,  it  is  proved  for  the  angle  at  d;  it  is  also 
true  for  any  other  angle ;  for,  since  all  other  inscribed  angles 
on  that  side  of  the  chord  line,  cf,  equal  the  angle  A,  (Art. 
161,)  therefore  the  angle  formed  by  a  chord  and  tangent  equals 
any  angle  in  the  opposite  segment  of  the  circle.  This  being 
proved  for  the  acute  angle,  Z>,  it  is  also  true  for  the  obtuse 


Fi*.  114. 

angle,  a  cf  ;  for,  from  any  point,  n,  (Fig.  114,)  in  the  arc 
c  nf,  draw  lines  to  d, /and  c;  now,  if  it  can  be  proved  t^at 
the  angle  a  of  equals  the  angle  f  n  c,  the  entire  proposition 
is  proved,  for  the  angle  f  n  c  equals  any  of  all  the  inscribed 
angles  that  can  be  drawn  on  that  side  of  the  chord.  (Art. 
161.)  To  prove,  then,  that  a  cf  equals  c  nf:  the  angle  a  cf 
equals  the  sum,  of  the  angles  A  and  B  /  also  the  angle  c  nf 
equals  the  sum  of  the  angles  C  and  D.  The  angles  B  and  D, 
being  inscribed  angles  on  the  same  chord,  df,  are  equal.  The 
angles  G  and  A  being  right  angles,  (Art.  156,)  are  likewise 
equal.  Now,  since  A  equals  (7,  and  B  equals  D,  therefore 
the  sum  of  A  and  B  equals  the  sum  of  C  and  D — or  the  angle 
a  cf  equals  the  angle  c  nf. 

164. — Proposition.  Two  chords,  a  5  and  c  d,  (Fig.  115,) 
intersecting,  the  parallelogram,  or  rectangle  formed  by  the  two 
parts  of  one  is  equal  to  the  rectangle  formed  by  the  two  parts 
of  the  other.  That  is,  c  e  mi  Itiplied  by  &  d,  the  product  ia 


72 


PRACTICAL   GEOMETRY. 


Fig.  116. 


equal  to  the  product  of  a  e  multiplied  by  e  5.  The  triangle  A 
is  similar  to  the  triangle  B,  because  it  has  corresponding  an- 
gles. The  angle  i  equals  the  angle  e,  (Art.  148  ;)  the  angle  at 
c  equals  the  angle  at  a  because  they  stand  upon  the  same 
chord,  d  5,  (Art.  161 ;)  for  the  same  reason  the  angle  b  equals 
the  angle  d,  for  each  stands  upon  the  same  chord,  a  c.  There- 
fore, the  triangle  A  having  the  same  angles  as  the  triangle  B, 
the  length  of  the  sides  of  one  are  in  like  proportion  as  the 
length  of  the  sides  in  the  other.  So,  ediaeileblce. 
Hence,  a  e  multiplied  by  e  5  is  equal  to  e  d  mutiplied  by  c  e — 
or  the  product  of  the  means  equals  the  product  of  the  ex- 
tremes. 

165. — Proposition.  In  any  circle,  when  a  segment  is  given, 
the  radius  is  equal  to  the  sum  of  the  squares  of  half  the  chord 
and  of  the  versed  sine,  divided  by  twice  the  versed  sine.  Let 
a  5,  (Fig.  116,)  be  the  chord  line,  and  v  the  versed  sine  of  the 
segment.  By  the  preceding  article  the  triangle  A  is  shown  to 
be  like  the  triangle  £,  having  equal  angles  and  proportionate 


AMERICAN   HOUSE-CARPENTEK. 


73 


Fig.  116. 

length  of  sides.    Therefore,  v  :  nr.m  :  i,  or  —  =  i  ;  that  is,  i  ia 

equal  to  the  square  of  n  (or  n  x  n)  divided  by  v.    This  result 
being  added  to  v  equals  the  diameter  o  #,  which  may  be  indi 

cated  by  the  letter  d j  thus, H  v  =i  +  v  =  d /  and  the  half 


of  this,  or- 


™  T 

-  =  —  —  r  =  the  radius.     Keducing  this  expres- 

a  2 

sion  by  multiplying  the  numerator  and  denominator  each  by  the 
like  quantity,  viz.  v,  there  results,  — =  r  /  and  where  c 


__ 

represents  the  chord,  the  expression  is,  ^-~ —  =  r:  that  is, 

as  stated  above,  the  radius  is  equal  to  the  sum  of  the  squares 
of  half  the  chord  and  of  the  versed  sine,  divided  by  twice  the 
versed  sine. 

166. — Proposition.    Any  ordinate,  m  n,  (Fig.  117,)  in  the 
segment  of  a  circle,  is  equal  to  the  square  root  of  the  difference 
10 


PRACTICAL   GEOMETRY. 


tig.  117. 


of  the  squares  of  the  radius  and  abscissa,  (d  n,}  less  the  differ- 
ence  of  the  radius  and  versed  sine.  So,  if  the  chord  a  5,  and 
the  versed  sine  c  d,  be  given,  the  length  of  any  number  of 
ordiuates  may  be  found  ~by  which  to  describe  the  arc.  Find 
the  radius,  c  e,  "by  the  preceding  Article.  It  will  be  observed 
that  e  m  is  also  radius.  Then,  to  find  the  length  of  the  ordi- 
nate,  m  n,  make  e  o  equal  to  d  n :  now,  according  to  Article 
157,  the  square  of  e  o  taken  from  the  square  of  e  m,  the  residue 
equals  the  square  of  o  m,  and  the  square  root  of  this  residue 
will  be  the  length  of  the  line  o  m.  Then  from  o  m  take  o  n 
equal  to  e  d,  and  the  result  will  be  the  length  of  m  n.  That  is, 
the  ordinate  is  equal  to  the  square  root  of  the  difference  of  the 
squares  of  the  radius  and  abscissa,  less  the  difference  of  the 
radius  and  versed  sine.  This  may  be  expressed  algebraically 
thus  :  y  =  ^/rt  —  a?  —  (r  —  v),  where  y  is  the  ordinate,  r  the 
radius,  x  the  abscissa,  and  v  the  versed  sine  ; — d  n  being  the 
abscissa  of  the  ordinate  nm^dg  the  abscissa  of  the  ordinate 


AMERICAN   HOUSE-CARPENTER.  75 

fff,  &c. :  the  abscissa  being  in  each  case  the  distance  from  the 
foot  of  the  versed  sine,  c  d,  to  the  foot  of  the  ordinate  whose 
length  is  sought. 


Fig.  118. 

167. — Proposition.  The  sides  of  any  quadrangle  being 
bisected,  and  lines  drawn  joining  the  points  of  bisection  in  the 
adjacent  sides,  these  lines  will  form  a  parallelogram.  Draw 
the  diagonals,  a  5  and  c  d,  (Fig.  118.)  It  will  here  be  per- 
ceived that  the  two  triangles,  a  e  o  and  a  c  d,  are  homologous, 
having  like  angles  and  proportionate  sides.  Two  of  the  sides 
of  one  triangle  lie  coincident  with  the  two  corresponding  sides 
of  the  other  triangle,  therefore  the  contained  angles  between 
these  sides  in  each  triangle  are  identical.  By  construction, 
these  corresponding  sides  are  proportionate ;  a  c  being  equal 
to  twice  a  0,  and  a  d  being  equal  to  twice  a  o  /  therefore  the 
remaining  sides  are  proportionate,  c  d  being  equal  to  twice  e  o, 
hence  the  remaining  corresponding  angles  are  equal.  Since, 
then,  the  angles  a  e  o  and  a  c  d  are  equal,  therefore  the  line  e  o 
is  parallel  with  the  diagonal  c  d — so,  likewise,  the  line  mnis 
parallel  to  the  same  diagonal,  c  d.  If,  therefore,  these  two 
lines,  e  o  and  m  n,  are  parallel  to  the  same  line,  c  d,  they  must 
be  parallel  to  each  other.  In  the  same  manner  the  lines  o  n 
and  e  m  are  proved  parallel  to  the  diagonal,  a  5,  and  to  each 


76  PRACTICAL   GEOMETRY. 

other ;  therefore  the  inscribed  figure,  m  e  o  n,  is  a  parallelo- 
gram. It  may  be  remarked  also,  that  the  parallelogram  so 
formed  will  contaii  just  one-half  the  area  of  the  circumscribing 
quadrangle. 


These  demonstrations,  which  relate  mostly  to  the  problems 
previously  given,  are  introduced  to  satisfy  the  learner  in  regard 
to  their  mathematical  accuracy.  By  studying  and  thoroughly 
understanding  them,  he  will  soonest  arrive  at  a  knowledge  of 
their  importance,  and  be  likely  the  longer  to  retain  them  in 
memory.  Should  he  have  a  relish  for  such  exercises,  and  wish 
to  continue  them  farther,  he  may  consult  Euclid's  Elements,  in 
which  the  whole  subject  of  theoretical  geometry  is  treated  of 
in  a  manner  sufficiently  intelligible  to  be  understood  by  the 
young  mechanic.  The  house-carpenter,  especially,  needs  infor- 
mation of  this  kind,  and  were  he  thoroughly  acquainted  with 
the  principles  of  geometry,  he  would  be  much  less  liable  to 
commit  mistakes,  and  be  better  qualified  to  excel  in  the  execu- 
tion of  liis  often  difficult  undertakings. 


SECTION  JI.— ARCHITECTURE. 


HISTORY    OF    ARCHITECTURE. 

168.— Architecture  has  been  defined  to  be — "  the  art  of  build 
ing  ;"  but,  in  it's  common  acceptation,  it  is — "  the  art  of  designing 
and  constructing  buildings,  in  accordance  with  such  principles  as 
constitute  stability,  utility  and  beauty."  The  literal  signification 
of  the  Greek  word  archi-tecton,  from  which  the  word  architect 
is  derived,  is  chief-carpenter ;  but  the  architect  has  always  been 
known  as  the  chief  designer  rather  than  the  chief  builder.  Of 
the  three  classes  into  which  architecture  has  been  divided — viz., 
Civil,  Military,  and  Naval,  the  first  is  that  which  refers  to  the 
construction  of  edifices  known  as  dwellings,  churches  and  other 
public  buildings,  bridges,  &c.,  for  the  accommodation  of  civilized 
man — and  is  the  subject  of  the  remarks  which  follow. 

169. — This  is  one  of  the  most  ancient  of  the  arts :  the  scrip- 
tures inform  us  of  its  existence  at  a  very  early  period.  Cain, 
the  son  of  Adam, — "builded-  a  city,  and  called  the  name  of  the 
city  after  the  name  of  his  son,  Enoch" — but  of  the  peculiar  style 
or  manner  of  building  we  are  not  informed.  It  is  presumed  that 
it  was  not  remarkable  for  beauty,  but  that  utility  and  perhaps  sta- 
bility were  its  characteristics.  Soon  after  the  deluge — that  me 


78  AMERICAN    HOUSE-CARPENTER. 

morable  event,  which  removed  from  existence  all  traces  of  the 
works  of  man — the  Tower  of  Babel  was  commenced.  This  was 
a  work  of  such  magnitude  that  the  gathering  of  the  materials, 
according  to  some  writers,  occupied  three  years  ;  the  period  from 
its  commencement  until  the  work  was  abandoned,  was  twenty- 
two  years ;  and  the  bricks  were  like  blocks  of  stone,  being  twenty 
feet  long,  fifteen  broad  and  seven  thick.  Learned  men  have  given 
it  as  their  opinion,  that  the  tower  in  the  temple  of  Belus  at  Baby 
Ion  was  the  same  as  that  which  in  the  scriptures  is  called  the 
Tower  of  Babel.  The  tower  of  the  temple  of  Belns  was  square 
at  its  base,  eacn  side  measuring  one  lurlong,  arid  consequently 
half  a  mile  in  circumference.  Its  form  was  that  of  a  pyramid 
and  its  height  was  660  feet.  It  had  a  winding  passage  on  the 
outside  from  the  base  to  the  summit,  which  was  wide  enough  for 
two  carriages. 

170. — Historical  accounts  of  ancient  cities,  of  which  there  are 
iiow  but  few  remains — such  as  Babylon,  Palmyra  and  Ninevah 
of  the  Assyrians ;  Sidon,  Tyre,  Aradus  and  Serepta  of  the  Phoe- 
nicians ;  and  Jerusalem,  with  its  splendid  temple,  of  the  Israelites 
— show  that  architecture  among  them  had  made  great  advances. 
Ancient  monuments  of  the  art  are  found  also  among  other  nations  j 
the  subterraneous  temples  of  the  Hindoos  upon  the  islands,  Ele- 
phanta  and  Salsetta ;  the  ruins  of  Persepolis  in  Persia ;  pyramids, 
obelisks,  temples,  palaces  and  sepulchres  in  Egypt — all  prove  that 
the  architects  of  those  early  times  were  possessed  of  skill  and 
judgment  highly  cultivated.  The  principal  characteristics  of 
their  works,  are  gigantic  dimensions,  immoveable  solidity,  and,  in 
some  instances,  harmonious  splendour.  The  extraordinary  size 
of  some  is  illustrated  in  the  pyramids  of  Egypt.  The  largest  of 
these  stands  not  far  from  the  city  of  Cairo :  its  base,  which  is 
square,  covers  about  11}  acres,  and  its  height  is  nearly  500  feet 
The  stones  of  which  it  is  built  are  immense — the  smallest  being 
full  thirty  feet  long. 

171.-  -Among  the  Greeks,  architecture  was  cultivated  as  a  fine 


ARCHITECTURE.  79 

art,  and  rapidly  advanced  towards  perfection.  Dignity  and  grace 
were  added  to  stability  and  magnificence.  In  the  Doric  order, 
their  first  style  of  building,  this  is  fully  exemplified.  Phidias. 
Ictinus  and  Callicrates,  are  spoken  of  as  masters  in  the  art  at  this 
period :  the  encouragement  and  support  of  Pericles  stimulated 
them  to  a  noble  emulation.  The  beautiful  temple  of  Minerva, 
erected  upon  the  acropolis  of  Athens,  the  Propyleum,  the  Odeum 
and  others,  were  lasting  monuments  of  their  success.  The  Ionic 
and  Corinthian  orders  were  added  to  the  Doric,  and  many  mag- 
nificent edifices  arose.  These  exemplified,  in  their  chaste  propor- 
tions, the  elegant  refinement  of  Grecian  taste.  Improvement  in 
Grecian  architecture  continued  to  advance,  until  perfection  seems 
to  have  been  attained.  The  specimens  which  have  been  partially 
preserved,  exhibit  a  combination  of  elegant  proportion,  dignified 
simplicity  and  majestic  grandeur.  Architecture  among  the 
Greeks  was  at  the  height  of  its  glory  at  the  period  immediately 
preceding  the  Peloponnesian  war;  after  which  the  art  declined. 
An  excess  of  enrichment  succeeded  its  former  simple  grandeur ; 
yet  a  strict  regularity  was  maintained  amid  the  profusion  of  orna- 
ment. After  the  death  of  Alexander,  323  B.  C.,  a  love  of  gaudy 
splendour  increased :  the  consequent  decline  of  the  art  was 
visible,  and  the  Greeks  afterwards  paid  but  little  attention  to  the 
science. 

172. — While  the  Greeks  were  masters  in  architecture,  which 
they  applied  mostly  to  their  temples  and  other  public  buildings, 
the  Romans  gave  their  attention  to  the  science  in  the  construction 
of  the  many  aqueducts  and  sewers  with  which  Rome  abounded ; 
building  no  such  splendid  edifices  as  adorned  Athens.  Corinth 
and  Ephesus.  until  about  200  years  B.  C.,  when  their  intercourse 
with  the  Greeks  became  more  extended.  Grecian  architecture 
was  introduced  into  Rome  by  Sylla ;  by  whom,  as  also  by  Marius 
and  Cassar,  many  large  edifices  were  erected  in  various  cities  of 
Italy.  But  under  Caesar  Augustus,  at  about  the  beginning  of  the 
Christian  era,  the  art  arose  to  the  greatest  perfection  it  ever  at- 


80  AMERICAN    HOUSE-CARPENTER. 

tained  in  Italy.  Under  his  patronage,  Grecian  artists  were  en- 
couraged, and  many  emigrated  to  Rome.  It  was  at  about  this 
time  that  Solomon's  temple  at  Jerusalem  was  rebuilt  by  Herod — 
a  Roman.  This  was  46  years  in  the  erection,  and  was  most  pro 
bably  of  the  Grecian  style  of  building — perhaps  of  the  Corin- 
thian order.  Some  of  the  stones  of  which  it  was  built  were  46 
feet  long,  21  feet  high  and  14  thick ;  and  others  were  of  the 
astonishing  length  of  82  feet,  The  porch  rose  to  a  great  height ; 
the  whole  being  built  of  white  marble  exquisitely  polished.  This 
is  the  building  concerning  which  it  was  remarked — "  Master,  see 
what  manner  of  stones,  and  what  buildings  are  here."  For  the 
construction  of  private  habitations  also,  finished  artists  were  em- 
ployed by  the  Romans  :  their  dwellings  being  often  built  with  the 
finest  marble,  and  their  villas  splendidly  adorned.  After  Augus- 
tus, his  successors  continued  to*  beautify  the  city,  until  the  reign  of 
Constantine ;  who,  having  removed  the  imperial  residence  to 
Constantinople,  neglected  to  add  to  the  splendour  of  Rome  ;  and 
the  art,  in  consequence,  soon  fell  from  its  high  excellence. 

Thus  we  find  that  Rome  was  indebted  to  Greece  for  what  she 
possessed  of  architecture — ;not  only  for  the  knowledge  of  its  prin- 
ciples, but  also  for  many  of  the  best  buildings  themselves  ;  these 
having  been  originally  erected  in  Greece,  and  stolen  by  the  un- 
principled conquerors — taken  down  and  removed  to  Rome. 
Greece  was  thus  robbed  of  her  best  monuments  of  architecture. 
Touched  by  the  Romans,  Grecian  architecture  lost  much  of  its 
elegance  and  dignity.  The  Romans,  though  justly  celebrated 
for  their  scientific  knowledge  as  displayed  in  the  construction  of 
their  various  edifices,  were  not  capable  of  appreciating  the  simple 
grandeur,  the  refined  elegance  of  the  Grecian  style ;  but  sought 
to  improve  upon  it  by  the  addition  of  luxurious  enrichment,  and 
thus  deprived  it  of  true  elegance.  In  the  days  of  Nero,  whose 
palace  of  gold  is  so  celebrated,  buildings  were  lavishly  adorned. 
Adrian  did  much  to  encourage  the  art ;  but  not  satisfied  with  the 
simplicity  of  the  Grecian  style,  the  artists  of  his  time  aimed  at 


ARCHITECTURE.  81 

inventing  rievr  ones,  and  added  to  the  already  redundant  embel- 
lishments of  the  previous  age.  Hence  the  origin  of  the  pedestal, 
the  great  variety  of  intricate  ornaments,  the  convex  frieze,  the 
round  and  the  open  pediments,  &c.  The  rage  for  luxury 
continued  until  Alexander  Severus,  who  made  some  improve- 
ment :  but  very  soon  after  his  reign,  the  art  began  rapidly  to 
decline,  as  particularly  evidenced  in  the  mean  and  trifling  charac- 
ter of  the  ornaments. 

173. — The  Goths  and  Vandals,  when  they  overran  the  coun- 
tries of  Italy,  Greece,  Asia  and  Africa,  destroyed  most  of  the 
works  of  ancient  architecture.  Cultivating  no  art  but  that  of 
war,  these  savage  hordes  could  not  be  expected  to  take  any  interest 
in  the  beautiful  forms  and  proportions  of  their  habitations.  From 
this  time,  architecture  assumed  an  entirely  different  aspect.  The 
celebrated  styles  of  Greece  were  unappreciated  and  forgotten;  and 
modern  architecture  took  its  first  step  on  the  platform  of  existence. 
The  Goths,  in  their  conquering  invasions,  gradually  extended  it 
over  Italy,  France,  Spain,  Portugal  and  Germany,  into  England. 
From  the  reign  of  Gallienus  may  be  reckoned  the  total  extinction 
of  the  arts  among  the  Romans.  From  his  time  until  the  6th  or 
7th  century,  architecture  was  almost  entirely  neglected.  The 
buildings  which  were  erected  during  this  suspension  of  the  arts, 
were  very  rude.  Being  constructed  of  the  fragments  of  the  edi- 
fices which  had  been  demolished  by  the  Visigoths  in  their  unre- 
strained fury,  and  the  builders  being  destitute  of  a  proper  know- 
ledge of  architecture,  many  sad  blunders  and  extensive  patch- 
work might  have  been  seen  in  their  construction— entablatures 
inverted,  columns  standing  on  their  wrong  ends,  and  other  ridi- 
culous arrangements  characterized  their  clumsy  work.  The  vast 
number  of  columns  which  the  ruins  around  them  afforded,  they 
used  as  piers  in  the  construction  of  arcades — which  by  some  is 
thought,  after  having  passed  through  various  changes,  to  hav; 
been  the  origin  of  the  plan  of  the  Gothic,  cathedral.  Buildings 
generally,  which  an  not  of  the  classical  styles,  and  which  were 
11 


82  AMERICAN    HOUSE-CARPENTER. 

erected  after  the  fall  of  the  Roman  empire,  have  by  some  been 
indiscriminately  included  under  the  term  Gothic.  But  the 
changes  which  architecture  underwent  during  the  dark  ages,  show 
that  there  were  several  distinct  modes  of  building. 

174. — Theodoric,  king  of  the  Ostrogoths,  a  friend  of  the  arts, 
who  reigned  in  Italy  from  A.  D.  493  to  525,  endeavoured  to  re- 
store and  preserve  some  of  the  ancient  buildings  ;  and  erected 
others,  the  ruins  of  which  are  still  seen  at  Verona  and  Ravenna. 
Simplicity  and  strength  are  the  characteristics  of  the  structures 
erected  by  him ;  they  are,  however,  devoid  of  grandeur  and  ele- 
gance, or  fine  proportions.  These  are  properly  of  the  GOTHIC 
style ;  by  some  called  the  old  Gothic  to  distinguish  it  from  the 
pointed  style,  which  is  generally  called  modern  Gothic. 

175. — The  Lombards,  who  ruled  in  Italy  from  A.  D.  568,  had 
no  taste  for  architecture  nor  respect  for  antiquities.  Accordingly, 
they  pulled  down  the  splendid  monuments  of  Classic  architecture 
which  they  found  standing,  and  erected  in  their  stead  huge  build- 
ings of  stone  which  were  greatly  destitute  of  proportion,  elegance 
or  utility — their  characteristics  being  scarcely  anything  more  than 
stability  and  immensity  combined  with  ornaments  of  a  puerile  cha- 
racter. Their  churches  were  disfigured  with  rows  of  small  columns 
along  the  cornice  of  the  pediment,  small  doors  and  windows  with 
circular  heads,  roofs  supported  by  arches  having  arched  buttresses 
to  resist  their  thrust,  and  a  lavish  display  of  incongruous  orna- 
ments. This  kind  of  architecture  is  called,  the  LOMBARD  style, 
and  was  employed  in  the  7th  century  in  Pavia,  the  chief  city  of 
the  Lombards;  at  which  city,  as  also  at  many  other  places,  a 
great  many  edifices  were  erected  in  accordance  with  its  inelegant 
forms.  , 

176. — The  Byzantine  architects,  from  Byzantium,  Constantino- 
ple, erected  many  spacious  edifices ;  among  which  are  included 
the  cathedra.s  of  Barnberg,  Worms  and  Mentz,  and  the  most  an 
cient  part  of  the  minster  at  Strasburg  ;  in  all  of  these  they  com- 
bined the  Roman-Ionic  order  with  the  Gothic  of  the  Lombards. 


ARCHITECTURE.  83 

This  style  is  called  the  LOMBARD-BYZANTINE.  To  the  last  style 
there  were  afterwards  added  cupolas  similar  to  thost  used  in  the 
east,  together  with  numerous  slender  pillars  with  tasteless  capi- 
tals, and  the  many  minarets  which  are  the  characteristics  of  the 
proper  Byzantine,  or  Oriental  style. 

177. — In  the  eighth  century,  when  the  Arabs  and  Moors  de- 
stroyed the  kingdom  of  the  Goths,  the  arts  and  sciences  were 
mostly  in  possession  of  the  Musselmen-conquerors ;  at  which 
time  there  were  three  kinds  of  architecture  practised ;  viz  :  the 
Arabian,  the  Moorish  and  the  modern-Gothic.  The  ARABIAN 
style  was  formed  from  Greek  models,  having  circular  arches 
added,  and  towers  which  terminated  with  globes  and  minarets. 
The  MOORISH  is  very  similar  to  the  Arabian,  being  distinguished 
from  it  by  arches  in  the  form  of  a  horse-shoe.  It  originated  in 
Spain  in  the  erection  of  buildings  with  the  ruins  of  Roman  archi- 
tecture, and  is  seen  in  all  its  splendour  in  the  ancient  palace  of  the 
Mohammedan  monarchs  at  Grenada,  called  the  Alhambra,  or  red- 
house.  The  MODERN-GOTHIC  was  originated  by  the  Visigoths 
in  Spain  by  a  combination  of  the  Arabian  and  Moorish  styles; 
and  introduced  by  Charlemagne  into  Germany.  On  account  of 
the  changes  and  improvements  it  there  underwent,  it  was,  at  about 
the  13th  or  14th  century,  termed  the  German,  or  romantic  style. 
It  is  exhibited  in  great  perfection  in  the  towers  of  the  minster  of 
Strasburgh,  the  cathedral  of  Cologne  and  other  edifices.  The 
most  remarkable  features  of  this  lofty  and  aspiring  style,  are  the 
lancet  or  pointed  arch,  clustered  pillars,  lofty  towers  and  flying 
buttresses.  It  was  principally  employed  in  ecclesiastical  archi- 
tecture, and  in  this  capacity  introduced  into  France,  Italy,  Spain, 
and  England. 

178. — The  Gothic  architecture  of  England  is  divided  into  the 
Norman,  the  Early-English,  the  Decorated,  and  the  Perpen- 
dicular styles.  The  Norman  is  principally  distinguished  by  the 
character  of  its  ornaments — the  chevron,  or  zigzag,  being  the 
most  common.  Buildings  in  this  style  were  erected  in  the  12tb 


84-  AMERICAN    HOUSE-CARPENTER. 

century.  The  Early-English  is  celebrated  for  the  beauty  of  its 
edifices,  the  chaste  simplicity  and  purity  of  design  which  they 
display,  and  the  peculiarly  graceful  character  of  its  foliage.  This 
style  is  of  the  IHth  century.  The  Decorated  style,  as  its  name 
implies,  is  characterized  by  a  great  profusion  of  enrichment, 
which  consists  principally  of  the  crocket,  or  feathered-ornament, 
and  ball-flower.  It  was  mostly  in  use  in  the  14th  century.  The 
Perpendicular  style,  which  dates  from  the  15th  century,  is  distin- 
guished by  its  high  towers,  and  parapets  surmounted  with  spires 
similar  in  number  and  grouping  to  oriental  minarets. 

179. — Thus  these  several  styles,  .which  have  been  erroneously 
termed  Gothic,  \veredistinguishedbypeculiarcharacteristicsaswell 
as  by  different  names.  The  first  symptoms  of  a  desire  to  return  to  a 
pure  style  in  architecture,  after  the  ruin  caused  by  the  Goths,  was 
manifested  in  the  character  of  the  art  as  displayed  in  the  church 
of  St.  Sophia  at  Constantinople,  which  was  erected  by  Justinian 
in  the  6th  century.  The  church  of  St.  Mark  at  Venice,  which 
arose  in  the  10th  or  llth  century,  was  the  work  of  Grecian  archi- 
tects, and  resembles  in  magnificence  the  forms  of  ancient  archi- 
tecture. The  cathedral  at  Pisa,  a  wonderful  structure  for  the  age, 
was  erected  by  a  Grecian  architect  in  1016.  The  marble  with 
which  the  walls  of  this  building  were  faced,  and  of  which  the  four 
rows  of  columns  that  support  the  roof  are  composed,  is  said  to  be 
of  an  excellent  character.  The  Campanile,  or  leaning-tower  as  it 
is  usually  called,  was  erected  near  the  cathedral  in  the  12th  cen- 
tury. Its  inclination  is  generally  supposed  to  have  arisen  from 
a  poor  foundation  ;  although  by  some  it  is  said  to  have  been  thus 
constructed  originally,  in  order  to  inspire  in  the  minds  of  the 
beholder  sensations  of  sublimity  and  awe.  In  the  13th  century, 
the  science  in  Italy  was  slowly  progressing ;  many  fine  churches 
were  erected,  the  style  of  which  displayed  a  decided  advance  in 
the  progress  towards  pure  classical  architecture.  In  other  parts 
of  Europe,  the  Gothic,  or  pointed  style,  was  prevalent.  The 
cathedral  at  Strasburg,  designed  *  y  Irwin  Steinbeck,  was  erected 


ARCHITECTURE.  85 

in  the  13th  and  14th  centuries.  In  France  and  England  dur- 
ing the  14th  century,  many  very  superior  edifices  were  erected 
in  this  style. 

180. — In  the  14th  and  15th  centuries,  and  particularly  in  the 
latter,  architecture  in  Italy  was  greatly  revived.  The  masters 
began  to  study  the  remains  of  ancient  Roman  edifices ;  and  many 
splendid  buildings  were  erected,  which  displayed  a  purer  taste 
in  the  science.  Among  others,  St.  Peter's  of  Eome,  which  was 
built  about  this  time,  is  a  lasting  monument  of  the  architectural 
skill  of  the  age.  Giocondo,  Michael  Angelo,  Palladio,  Yiguola, 
and  other  celebrated  architects,  each  in  their  turn,  did  much  to 
restore  the  art  to  its  former  excellence.  In  the  edifices  which 
were  erected  under  their  direction,  however,  it  is  plainly  to  be 
seen  that  they  studied  not  from  the  pure  models  of  Greece,  but 
from  the  remains  of  the  deteriorated  architecture  of  Rome.  The 
high  pedestal,  the  coupled  columns,  the  rounded  pediment,  the 
many  curved-and-twisted  enrichments,  and  the  convex  frieze, 
were  unknown  to  pure  Grecian  architecture.  Yet  their  efforts 
were  serviceable  in  correcting,  to  a  good  degree,  the  very 
impure  taste  that  had  prevailed  since  the  overthrow  of  the  Ro- 
man empire. 

181. — At  about  this  time,  the  Italian  masters  and  numerous 
artists  who  had  visited  Italy  for  the  purpose,  spread  the  Roman 
style  over  various  countries  of  Europe  ;  which  was  gradually  re- 
ceived into  favor  in  place  of  the  modern-Gothic.  This  fell  into 
disuse  ;  although  it  has  of  late  years  been  again  cultivated.  It 
requires  a  building  of  great  magnitude  and  complexity  for  a  per- 
fect display  of  its  beauties.  In  America,  the  pure  Grecian  style 
was  at  first  more  or  less  studied ;  and  perhaps  the  simplicity  of 
its  principles  would  be  better  adapted  to  a  republican  country, 
than  the  intricacy  and  extent  of  those  of  the  Gothic ;  but  at  the 
present  time  the  latter  style  is  being  introduced,  especially  for 
ecclesiastical  structures. 


86  AMERICAN    HOUSE-CARPENTER. 

STYLES   OF   ARCHITECTURE. 

152. It  is  generally  acknowledged  that  the  various  styles  in 

architecture,  were  originated  in  accordance  with  the  different 
pursuits  of  the  early  inhabitants  of  the  earth ;  and  were  brought 
by  their  descendants  to  their  present  state  of  perfection,  through 
the  propensity  for  imitation  and  desire  of  emulation  which  are 
found  more  or  less  among  all  nations.  Those  that  followed 
agricultural  pursuits,  from  being  employed  constantly  upon 
the  same  piece  of  land,  needed  a  permanent  residence,  and  the 
wooden  hut  was  the  offspring  of  their  wants ;  while  the  shep- 
herd, who  followed  his  flocks  and  was  compelled  to  traverse 
large  tracts  of  country  for  pasture,  found  the  tent  to  be  the 
most  portable  habitation ;  again,  the  man  devoted  to  hunting 
and  fishing — an  idle  and  vagabond  way  of  living — is  naturally 
supposed  to  have  been  content  with  the  cavern  as  a  place  of 
shelter.  The  latter  is  said  to  have  been  the  origin  of  the 
Egyptian  style ;  while  the  curved  roof  of.  Chinese  structures 
gives  a  strong  indication  of  their  having  had  the  tent  for  their 
model ;  and  the  simplicity  of  the  original  style  of  the  Greeks, 
(the  Doric,)  shows  quite  conclusively,  as  is  generally  conceded, 
that  its  original  was  of  wood.  The  modern-Gothic,  or  pointed 
style,  which  was  most  generally  confined  to  ecclesiastical 
structures,  is  said  by  some  to  have  originated  in  an  attempt  to 
imitate  the  bower,  or  grove  of  trees,  in  which  the  ancients  per- 
formed their  idol-worship. 

183. — There  are  numerous  styles,  or  orders,  in  architecture  ; 
and  a  knowledge  of  the  peculiarities  of  each  is  important  to  the 
student  in  the  art.  An  ORDER,  in  architecture,  is  composed  of 
three  principal  parts,  viz :  the  Stylobate,  the  Column  and  the 
Entablature. 

184. — The  STYLOBATE  is  the  substructure,  or  basement,  upon 
which  the  columns  of  an  order  are  arranged.  In  Koman  archi- 
tecture— especially  in  the  interior  of  an  edifice — it  frequently 
occursjjhat  each  column  has  a  separate  substructure ;  this  is 


ARCHITECTURE.  87 

called  &  pedestal.  If  possible,  the  pedestal  should  be  avoided 
in  all  cases ;  because  it  gives  to  the  column,  the  appearance  of 
having  been  originally  designed  for  a  small  building,  and  after- 
wards pieced-out  to  make  it  long  enough  for  a  larger  one. 

185. — The  COLUMN  is  composed  of  the  base,  shaft  and  capital. 

186. — The  ENTABLATURE,  above  and  supported  by  the  co- 
lumns, is  horizontal ;  and  is  composed  of  the  architrave,  frieze 
and  cornice.  These  principal  parts  are  again  divided  into 
vavious  members  and  mouldings.  (See  Sect.  III.) 

187. — The  BASE  of  a  column  is  so  called  from  lasis,  a  found- 
ation, or  footing. 

•188. — The  SHAFT,  the  upright  part  of  a  column  standing 
upon  the  base  and  crowned  with  the  capital,  is  from  shafto,  to 
dig — in  the  manner  of  a  well,  whose  inside  is  not  unlike  the 
form  of  a  column. 

189. — The  CAPITAL,  from  "kepliale  or  caput,  the  head,  is  the 
uppermost  and  crowning  part  of  the  column. 

190. — The  ARCHITRAVE,  from  archi,  chief  or  principal,  and 
trahs,  a  beam,  is  that  part  of  the  entablature  which  lies  in 
immediate  connection  with  the  column. 

191, — The  FKIEZE,  from  fibron,  a  fringe  or  border,  is  that 
part  of  the  entablature  which  is  immediately  above  the  archi- 
trave and  beneath  the  cornice.  It  was  called  by  some  of  the 
ancients,  zophorus,  because  it  was  usually  enriched  with  sculp- 
tured animals. 

192. — The  CORNICE,  from  corona,  a  crown,  is  the  upper  and 
projecting  part  of  the  entablature — being  also  the  uppermost 
and  crowning  part  of  the  whole  order. 

193. — The  PEDIMENT,  above  the  entablature,  is  the  triangular 
portion  which  is  formed  by  the  inclined  edges  of  the  roof  at 
the  end  of  the  building.  In  Gothic  architecture,  the  pediment 
is  called,  a  gable. 

19-i. — The  TYMPANUM  is  the  perpendicular  triangular  surface 
which  is  enclosed  by  the  c«  mice  of  the  pediment. 


83  AMERICAN   HOUSE-CARPENTER. 

195. The  zVrnc  is  a  small  order,  consisting  of  pilasters  and 

entablature,  raised  above  a  hrger  order,  instead  of  a  pediment 
An  attic  story  is  the  upper  story,  its  windows  being  usually 
square. 

196. — An  order,  in  architecture,  has  its  several  parts  and 
members  proportioned  to  one  another  by  a  scale  of  60  equal 
parts,  which  are  called  minutes.  If  the  height  of  buildings 
were  always  the  same,  the  scale  of  equal  parts  would  be  a 
fixed  quantity — an  exact  number  of  feet  and  inches.  But  as 
buildings  are  erected  of  different  heights,  the  column  and  its 
accompaniments  are  required  to  be  of  different  dimensions. 
To  ascertain  the  scale  of  equal  parts,  it  is  necessary  to  knot?' 
the  height  to  which  the  whole  order  is  to  be  erected.  This 
must  be  divided  by  the  number  of  diameters  which  is  directed 
for  the  order  under  consideration.  Then  the  quotient  obtained 
by  such  division,  is  the  length  of  the  scale  of  equal  parts — and 
is,  also,  the  diameter  of  the  column  next  above  the  base.  For 
instance,  in  the  Grecian  Doric  order  the  whole  height,  includ- 
ing column  and  entablature,  is  8  diameters.  Suppose  now  it 
were  desirable  to  construct  an  example  of  this  order,  forty  feet 
high.  Then  40  feet  divided  by  8,  gives  5  feet  for  the  length 
of  the  scale ;  and  this  being  divided  by  60,  the  scale  is  com- 
pleted. The  upright- columns  of  figures,  marked  Hand  P,  by 
the  side  of  the  drawings  illustrating  the  orders,  designate  the 
height  and  the  projection  of  the  members.  The  projection  of 
each  member  is  reckoned  from  a  line  passing  through  the  axis 
of  the  column,  and  extending  above  it  to  the  top  of  the  enta- 
blature. The  figures  represent  minutes,  or  60ths,  of  the  major 
diameter  of  the  shaft  of  the  column. 

197. — GRECIAN  STYLES.  The  original  method  of  building 
among  the  Greeks,  was  in  what  is  called  the  Doric  order :  to 
this  were  afterwards  added  the  Ionic  and  the  Corinthian. 
These  three  were  the  only  styles  known  among  them.  Each  is 
distinguish!  d  from  the  other  two,  by  not  only  a  peculiarity  of 


ARCHITECTURE.  89 

some  one  or  more  of  its  principal  parts,  but  also  by  a  particular 
destination.  The  character  of  the  Doric  is  robust,  manly  and 
Herculean-like ;  that  of  the  Ionic  is  more  delicate,  feminine, 
matronly ;  while  that  of  the  Corinthian  is  extremely  delicate, 
youthful  and  virgin-like.  However  they  may  differ  in  theii 
general  character,  they  are  alike  famous  for  grace  and  dignity, 
elegance  and  grandeur,  to  a  high  degree  of  perfection. 

198. — The  DORIC  ORDER,  (Fig.  120,)  is  so  ancient  that  its  origin 
is  unknown — although  some  have  pretended  to  have  discovered 
it.  But  the  most  general  opinion  is,  that  it  is  an  improvement 
upon  the  original  wooden  buildings  of  the  Grecians.  These  no 
doubt  were  very  rude,  and  perhaps  not  unlike  the  following 
figure. 


Fig.  119. 

The  trunks  of  trees,  set  perpendicularly  to  support  the  roof, 
may  be  taken  for  columns  ;  the  tree  laid  upon  the  tops  of  the 
perpendicular  ones,  the  architrave  ;  the  ends  of  the  cross-beams 
which  rest  upon  the  architrave,  the  triglyphs ;  the  tree  laid  on 
the  cross-beams  as  a  support  for  the  ends  of  the  rafters,  the 
bed-moulding  of  the  cornice ;  the  ends  of  the  rafters  which 
project  beyond  the  bed-moulding,  the  mutules;  and  perhaps 
the  projection  of  the  roof  in  front,  to  screen  the  entrance  from 
the  weather,  gave  origin  to  the  portico. 

The  peculiarities  of  the  Doric  order  are  the  triglyphs — those 

parts  of  the  frieze  which  have  perpendicular  channels  cut  in 
12 


DORIC  ORDER. 


Fig.  120 


ARCHITECTURE.  91 

tneir  surface  ;  the  absence  of  a  base  to  the  column — as  alfo  of 
fillets  between  the  flutings  of  the  column,  and  the  plainness  of 
the  capital.  The  triglyphs  are  to  be  so  disposed  that  the  width 
of  the  metopes — the  spaces  between  the  triglyphs — shall  be 
equal  to  their  height. 

199. — The  intercolumniation,  or  space  between  the  columns, 
is  regulated  by  placing  the  centres  of  the  columns  under  the 
centres  of  the  triglyphs — except  at  the  angle  of  the  building ; 
where,  as  may  be  seen  in  Fig.  120,  one  edge  of  the  triglyph 
must  be  over  the  centre  of  the  column.*  Where  the  columns 
are  so  disposed  that  one  of  them  stands  beneath  every  other  tri- 
glyph, the  arrangement  is  caUed,  mono-triglypJi,  and  is  most 
common.  When  a  column  is  placed  beneath  every  third  tri- 
glyph, the  arrangement  is  called  diastyle  j  and  when  beneath 
every  fourth,  arceostyle.  This  last  style  is  the  worst,  and  is  sel- 
dom adopted. 

200. — The  Doric  order  is  suitable  for  buildings  that  are  des- 
tined for  national  purposes,  for  banking-houses,  &c.  Its  ap- 
pearance, though  massive  and  grand,  is  nevertheless  rich  and 
graceful.  The  Patent  Office  at  Washington,«and  the  Custom- 
House  at  New  York,  are  good  specimens  of  this  order. 


*  GRECIAN  DORIC  ORDER.     When  the  width  to  be  occupied  by  the  whole  front  is  limited;  to  deter- 
mine the  diameter  of  the  column. 
The  relation  between  the  parts  may  be  expressed  thus: 


d(6  +  c)  +  (60  —  c) 

Where  a  equals  the  width  in  feet  occupied  by  the  columns,  and  their  intercolumniatjpns  taken 
collectively,  measured  at  the  base ;  *  equals  the  width  of  the  metope,  in  minutes ;  c  equals  the  width 
of  the  triclyphs  in  minutes ;  d  equals  the  number  of  metopes,  and  x  equals  the  diameter  in  feet. 

Example. — A  front  of  six  columns — hexastyle — 61  feet  wide;  the  frieze  having  one  triglyph  over 
each  intercolnmiiiation,  or  mono-triglyph.  In  this  case,  there  being  five  intercolummations  and  two 
metopes  over  each,  therefore  there  are  5  X  2  =  !0  metopes.  Let  the  metope  equal  42  minutes  and 
the  triglyph  equal  *}.  Then  a  —  61 ;  b  =  42 ;  c  =  28 ;  and  d  =  10 ;  and  the  formula  above  becomes, 

r_ 60X61*  6U  X  61      _3660_ 

I  —  28)  ~~  10  X  70  -f  32  ~  ?32~~ 


Example.— An  octastyle  front,  8  columns,  184  feet  wide,  three  metopes  over  each  intercolurani» 
Bon,  21  in  all,  and  the  metope  and  triglyph  42  ani.  26,  as  before.    Then, 
60  x  134  11040 

8«r~S8)  =  1502  =  V'35 r?ff a  feet  =  *e  diameter  ^quired. 


IONIC  ORDER. 


ARCHITECTURE.  93 

201.— The  IONIC  ORDER.  (Fig.  121.)  The  Doric  was  for 
some  time  the  only  order  in  use  among  the  Greeks.  They  gave 
their  attention  to  the  cultivation  of  it,  until  perfection  seems  to 
have  been  attained.  Their  temples  were  the  principal  objects 
upon  which  their  skill  in  the  art  was  displayed ;  and  as  the 
Doric  order  seems  to  have  been  well  fitted,  by  its  massive  pro- 
portions, to  represent  the  character  of  their  male  deities  rather 
than  the  female,  there  seems  to  have  been  a  necessity  for  an- 
other style  which  should  be  emblematical  of  feminine  graces, 
and  with  which  they  might  decorate  such  temples  as  were  de- 
dicated to  the  goddesses.  Hence  the  origin  of  the  Ionic  order. 
This  was  invented,  according  to  historians,  by  Hermogenes  of 
Alabanda ;  and  he  being  a  native  of  Caria,  then  in  the  posses- 
sion of  the  lonians,  the  order  was  called,  the  Ionic. 

202. — The  distinguishing  features  of  this  order  are  the  vo 
lutes,  or  spirals  of  the  capital ;  and  the  dentils  among  the  bed- 
mouldings  of  the  cornice :  although  in  some  instances,  dentils 
are  wanting.  The  volutes  are  said  to  have  been  designed  as  a 
representation  of  curls  of  hair  on  the  head  of  a  matron,  of  whom 
the  whole  column  is  taken  as  a  semblance. 

203. — The  intercolumniation  of  this  and  the  other  orders — 
both  Roman  and  Grecian,  with  the  exception  of  the  Doric- 
are  distinguished  as  follows.  When  the  interval  is  one  and  a 
half  diameters,  it  is  called,  pycnostyle,  or  columns  thick-set ; 
when  two  diameters,  sy style  /  when  two  and  a  quarter  diame- 
ters, eustyle  /  when  three  diameters,  diastyle  /  and  when  more 
than  three  diameters,  arceostyle,  or  columns  thin-set.  In  all  the 
orders,  when  there  are  four  columns  in  one  row,  the  arrange- 
ment is  called,  tetrastyle  $  when  there  are  six  in  a  row,  hexa- 
style  ;  and  when  eight,  octastyle. 

204. — The  Ionic  order  is  appropriate  for  churches,  colleges, 
seminaries,  libraries,  all  edifices  dedicated  to  literature  and  the 
arts,  and  all  places  of  peace  and  tranquillity.  The  front  of  the 


94: 


ARCHITECTURE- 


Merchants'  Exchange,  New  York  city,  is  a  good  spec im  en  of 
this  order. 

205. — To  describe  the  Ionic  volute.  Draw  a  perpendicular 
from  a  to  s,  (Fig.  122,)  and  make  a  8  equal  to  20  min.  or  to  4 
of  the  whole  height,  a  c  /  draw  s  0,  at  right  angles  to  s  a,  and 
equal  to  3  J  min. ;  upon  0,  with  2J  min.  for  radius,  describe  the 
eye  of  the  volute ;  about  0,  the  centre  of  the  eye,  draw  the 
square,  r  1 1  2,  with  sides  equal  to  half  the  diameter  of  the 
eye,  viz.  2$  min.,  and  divide  it  into  IM  equal  parts,  as  shown 


AMERICAN   HOUSE-CARPENTER. 


Fig.  123. 

at  Fig.  123.  The  several  centres  in  rotation  are  at  the  angles 
formed  by  the  heavy  lines,  as  figured,  1,  2,  3,  4,  5,  6,  &c.  The 
position  of  these  angles  is  determined  by  commencing  at  the 
point,  1,  and  making  each  heavy  line  one  part  less  in  length 
than  the  preceding  one.  No.  1  is  the  centre  for  .the  arc,  a  5, 
(Fig.  122 ;)  2  is  the  centre  for  the  arc,  I  c ;  and  so  on  to  the 
last.  The  inside  spiral  line  is  to  be  described  from  the  centres, 
a?,  a?,  x,  &c.,  (Fig.  123,)  being  the  centre  of  the  first  small 
square  towards  the  middle  of  the  eye  from  the  centre  for  the 
outside  arc.  The  breadth  of  the  fillet  at  a  j,  is  to  be  made 
equal  to  2T3F  min.  This  is  for  a  spiral  of  three  revolutions ;  but 
one  of  any  number  of  revolutions,  as  4  or  6,  may  be  drawn,  by 
dividing  of,  (Fig.  123,)  into  a  corresponding  number  of  equal 
parts.  Then  divide  the  part  nearest  the  centre,  0,  into  two 
parts,  as  at  h  j  join  o  and  1,  also  o  and  2;  draw  h  3,  parallel 
to  o  1,  and  h  4,  parallel  to  o  2 ;  then  the  lines,  o  1,  o  2,  h  3,  h  4, 
will  determine  the  length  of  the  heavy  lines,  and  the  place  of 
the  centres.  (See  Art.  489.) 


96  AMERICAN   HOUSE-CARPENTER. 

206.— The  CORINTHIAN  ORDER,  (Fig.  125,)  is  in  general  like 
the  Ionic,  though  the  proportions  are  lighter.  The  Corinthian 
displays  a  more  airy  elegance,  a  richer  appearance  ;  but  its 
distinguishing  feature  is  its  beautiful  capital.  This  is  gene- 
rally supposed  to  have  had  its  origin  in  the  capitals  of  the 
columns  of  Egyptian  temples  ;  which,  though  not  approaching 
it  in  elegance,  have  yet  a  similarity  of  form  with  the  Corin- 
thian. The  oft-repeated  story  of  its  origin  which  is  told  by 
Vitruvius— an  architect  who  flourished  in  Eome,  in  the  days 
of  Augustus  Caesar — though  pretty  generally  considered  to  be 
fabulous,  is  nevertheless  worthy  of  being  again  recited.  It  is 
this :  a  young  lady  of  Corinth  was  sick,  and  finally  died. 
Her  nurse  gathered  into  a  doep  basket,  such  trinkets  and 
keepsakes  as  the  lady  had  been  fond  of  when  alive,  and 
placed  them  upon  her  grave  ;  covering  the  basket  with  a  flat 
stone  or  tile,  that  its  contents  might  not  be  disturbed.  The 
basket  was  placed  accidentally  upon  the  stem  of  an  acanthus 
plant,  which,  shooting  forth,  enclosed  the  basket  with  its  foli- 
age ;  some  of  which,  reaching  the  tile,  turned  gracefully  over 
in  the  form  of  a  volute. 

A   celebrated    sculptor,    Calima- 
chus,  saw  the  basket  thus  deco- 
rated, and  from  the  hint  which  it 
suggested,    conceived    and    con- 
structed a -capital  for  a  column.  ' 
This  was  called  Corinthian  from 
the  fact  that  it  was  invented  and 
first  made  use  of  at  Corinth. 
207. — The  Corinthian  being  the  gayest,  the  richest,  and 
most  lovely  of  all  the  orders,  it  is  appropriate  for  edifices 
which  are  dedicated  to  amusement,  banqueting  and  festiv- 
ity— for  all  places  where  delicacy,  gayety  and  splendour  are 
desirable. 
208. — In  addition  to  the  three  regular  orders  of  architecture, 


ARCHITECTURE. 


T'T 


CORINTHIAN  ORDER.— Fig.  125. 

13 


98  AMERICAN    HOUSE-CARPENTER. 

it  was  sometimes  customary  among  the  Greeks — and  after 
wards  among  other  nations — to  employ  representations  of  the 
human  form,  instead  of  columns,  to  support  entablatures ;  these 
were  called  Persians  and  Caryatides. 

209. — PERSIANS  are  statues  of  men,  and  are  so  called  in 
commemoration  of  a  victory  gained  over  the  Persians  by  Pau- 
sanias.  The  Persian  prisoners  were  brought  to  Athens  and 
condemned  to  abject  slavery ;  and  in  order  to  represent  them 
in  the  lowest  state  of  servitude  and  degradation,  the  ^statues 
were  loaded  with  the  heaviest  entablature,  the  Doric. 

210. — CARYATIDES  are  statues  of  women  dressed  in  long 
robes  after  the  Asiatic  manner.  Their  origin  is  as  follows. 
In  a  war  between  the  Greeks  and  the  Caryans,  the  latter  were 
totally  vanquished,  their  male  population  extinguished,  and 
their  females  carried  to  Athens.  To  perpetuate  the  memory 
of  this  event,  statues  of  females,  having  the  form  and  dress  of 
the  Caryans,  were  erected,  and  crowned  with  the  Ionic  or  Co- 
rinthian entablature.  The  can^atides  were  generally  formed 
of  about  the  human  size,  but  the  persians  much  larger ;  in 
order  to  produce  the  greater  awe  and  astonishment  in  the 
beholder.  The  entablatures  were  proportioned  to  a  statue  in 
like  manner  as  to  a  column  of  the  same  height. 

211. — These  semblances  of  slavery  have  been  in  frequent 
use  among  moderns  as  well  as  ancients ;  and  as  a  relief  from 
the  stateliness  and  formality  of  tjie  regular  orders,  are  capable 
of  forming  a  thousand  varieties  ;  yet  in  a  land  of  liberty  such 
marks  of  human  degradation  ought  not  to  be  perpetuated. 

212. — ROMAN  STYLES.  Strictly  speaking,  Rome  had  no 
architecture  of  her  own — all  she  possessed  was  borrowed  from 
other  nations.  Before  the  Romans  exchanged  intercourse 
with  the  Greeks,  they  possessed  some  edifices  of  considerable 
extent  and  merit,  which  were  erected  by  architects  from  Etru- 
ria ;  but  Rome*  was  principally  indebted  to  Greece  for  what 
she  acquired  of  the  art.  Although  there  is  no  such  thing  as 


AECHITECTUKE. 


^JO  AMERICAN   HWSE-CARPEXTER. 

an  architecture  of  Roman  invention,  yet  no  nation,  perhaps, 
ever  was  so  devoted  to  the  cultivation  of  the  art  as  the  Ro- 
man. "Whether  we  consider  the  number  and  extent  of  their 
structures,  or  the  lavish  richness  and  splendour  with  which 
they  were  adorned,  we  are  compelled  to  yield  to  them  our 
admiration  and  praise.  At  one  time,  under  the  consuls  and 
emperors,  Rome  employed  400  architects.  The  public  works 
— such  as  theatres,  circuses,  baths,  aqueducts,  &c. — were,  in 
extent  and  grandeur,  beyond  any  thing  attempted  in  modern 
times.  Aqueducts  were  built  to  convey  water  from  a  distance 
of  60  miles  or  more.  In  the  prosecution  of  this  work,  rocks 
and  mountains  were  tunnelled,  and  valleys  bridged.  Some  of 
the  latter  descended  200  feet  below  the  level  of  the  water ; 
and  in  passing  them  the  canals  were  supported  by  an  arcade, 
or  succession  of  arches.  Public  baths  are  spoken  of  as  large 
as  cities ;  being  fitted  up  with  numerous  conveniences  for 
exercise  and  amusement.  Their  decorations  were  most  splen- 
did ;  indeed,  the  exuberance  of  the  ornaments  alone  was  offen- 
sive to  good  taste.  So  overloaded  with  enrichments  were  the 
baths  of  Diocletian,  that  on  an  occasion  of  public  festivity, 
great  quantities  of  sculpture  fell  from  the  ceilings  and  entabla- 
tures, killing  many  of  the  people. 

213. — The  three  orders  of  Greece  were  introduced  into 
Rome  in  all  the  richness  and  elegance  of  their  perfection. 
But  the  luxurious  Romans,  not  satisfied  with  the  simple  ele- 
gance of  their  refined  proportions,  sought  to  improve  upon 
them  by  lavish  displays  of  ornament.  They  transformed  in 
many  instances,  the  true  elegance  of  the  Grecian  art  into  a 
gaudy  splendour,  better  suited  to  their  less  refined  taste.  The 
Romans  remodelled  each  of  the  orders :  the  Doric,  (Fig.  126,) 
was  modified  by  increasing  the  heignt  of  the  column  to  8  dia- 
meters ;  by  changing  the  echinus  of  the  capital  for  an  ovolo, 
or  quarter  round,  and  adding  an  astragal  and  neck  below  it ; 
by  placing  the  centre,  instead  of  one  edge,  of  the  first  triglyph 


ARCHITECTURE. 


101 


m 


II I      I        I 


> 


102  AMERICAN   HOUSE-CAKPENTEB. 

ovei  the  centre  of  the  column;  and  introducing  horizontal 
instead  of  inclined  mutules  in  the  cornice,  and  in  some  instan 
ces  dispensing  with  them  altogether.  The  Ionic  was  modified 
by  diminishing  the  size  of  the  volutes,  and,  in  some  specimens, 
introducing  a  new  capital  in  which  the  volutes  were  diago- 
nally arranged,  (Fig.  127.)  This  new  capital  has  been  termed 
modern  Ionic.  The  favorite  order  at  Rome  and  her  colonies 
was  the  Corinthian,  (Fig.  128.)  But  this  order,  the  Roman 
artists  in  their  search  for  novelty,  subjected  to  many  altera- 
tions— especially  in  the  foliage  of  its  capital.  Into  the  upper 
part  of  this,  they  introduced  the  modified  Ionic  capital ;  thus 
combining  the  two  in  one.  This  change  was  dignified  with 
the  importance  of  an  order,  and  received  the  appellation, 
COMPOSITE,  or  Roman :  the  best  specimen  of  which  is  found  in 
the  Arch  of  Titus,  (Fig.  129.)  This  style  was  not  much  used 
among  the  Romans  themselves,  and  is  but  slightly  appreciated 
now. 

214. — The  TUSCAN  OKDER  is  said  to  have  been  introduced  to 
the  Romans  by  the  Etruscan  architects,  and  to  have  been  the 
only  style  used  in  Italy  before  the  introduction  of  the  Grecian 
orders.  However  this  may  be,  its  similarity  to  the  Doric 
order  gives  strong  indications  of  its  having  been  a  rude  imita- 
tion of  that  style :  this  is  very  probable,  since  history  informs 
us  that  the  Etruscans  held  intercourse  with  the  Greeks  at  a 
remote  period.  The  rudeness  of  this  order  prevented  its  ex- 
tensive use  in  Italy.  All  that  is  known  concerning  it  is  from 
Vitruvius — no  remains  of  buildings  in  this  style  being  found 
among  ancient  ruins. 

215. — For  mills,  factories,  markets,  barns,  stables,  &c.,  where 
utility  and  strength  are  of  more  importance  than  beauty,  the 
improved  modification  of  this  order,  called  the  modern  Tuscan, 
(Fig.  130,)  will  be  useful ;  and  its  simplicity  recommends  it 
where  economy  is  desirable. 

216. — EGYPTIAN  STYLE.    The   architecture   of  the   ancient 


ARCHITECTURE. 


103 


Fig.  128. 


104 


SSL 


AMERICAN   HOUSE-OARPENTEE. 


TUSCAN   ORDER. 


105 


H.  P. 


30   25 


l|     6 


10    39 

S 

10   25 
~3!"27 


c 


Fiar.  130. 

14 


106  ARCHITECTURE. 

Egyptians  -to  which  that  of  the  ancient  Hindoos  bears  some  re- 
semblance— is  characterized  by  boldness  of  outline,  solidity  and 
grandeur.  The  imozing  labyrinths  and  extensive  artificial  lakes, 
the  splendid  palaces  and  gloomy  cemeteries,  the  gigantic  pyramids 
and  towering  obelisks,  of  the  Egyptians,  were  works  of  immen- 
sity and  durability ;  and  their  extensive  remains  are  enduring 
proofs  of  the  enlightened  skill  of  this  once-powerful,  but  long  since 
extinct  nation.  The  principal  features  of  the  Egyptian  Style  of 
architecture  are — uniformity  of  plan,  never  deviating  from  right 
lines  and  angles ;  thick  walls,  having  the  outer  surface  slightly 
deviating  inwardly  from  the  perpendicular ;  the  whole  building 
low ;  roof  flat,  composed  of  stones  reaching  in  one  piece  from  pier 
to  pier,  these  being  supported  by  enormous  columns,  very  stout  in 
proportion  to  their  height;  the  shaft  sometimes  polygonal,  having 
no  base  but  with  a  great  variety  of  handsome  capitals,  the  foliage 
of  these  being  of  the  palm,  lotus  and  other  leaves ;  entablatures 
having  simply  an  architrave,  crowned  with  a  huge  cavetto  orna- 
mented with  sculpture  ;  and  the  intercolumniation  very  narrow, 
usually  U  diameters  and  seldom  exceeding  2£.  In  the  remains 
of  a  temple,  the  walls  were  found  to  be  24  feet  thick  ;  and  at  the 
gates  of  Thebes,  the  walls  at  the  foundation  were  50  feet  thick 
and  perfectly  solid.  The  immense  stones  of  which  these,  as  well 
as  Egyptian  walls  generally,  were  built,  had  both  their  inside  and 
outside  surfaces  faced,  and  the  joints  throughout  the  body  of  the 
wall  as  perfectly  close  as  upon  the  outer  surface.  For  this  reason, 
as  well  as  that  the  buildings  generally  partake  of  the  pyramidal 
form,  arise  their  great  solidity  and  durability.  The  dimensions 
and  extent  of  the  buildings  may  be  judged  from  the  temple  ol 
Jupiter  at  Thebes,  which  was  1400  feet  long  and  300  feet  wide  — 
exclusive  of  the  porticos,  of  which  there  was  a  great  number. 

It  is  estimated  by  Mr.  Gliddon,  U.  S.  consul  in  Egypt,  that  not 
less  than  25,000,000  tons  of  hewn  stone  were  employed  in  the 
erection  of  the  Pyramids  of  Memphis  alone, — or  enough  to  con- 
struct 3,000  Bunker-Hill  monuments.  Some  of  the  blocKs  are  40 


EGYPTIAN    STYLE. 


ic; 


H.  P. 


1*   55 


Fig.  131. 


10<  -  ARCHITECTURE. 

feet  long,  and  polished  with  emery  to  a  surprising  degree.  It  is 
conjectured  that  the  stone  for  these  pyramids  was  brought,  by 
rafts  and  canals,  from  a  distance  of  6  or  7  hundred  miles. 

217. — The  general  appearance  of  the  Egyptian  style  of  archi- 
tecture is  that  of  solemn  grandeur — amounting  sometimes  to 
sepulchral  gloom.  For  this  reason  it  is  appropriate  for  cemete- 
ries, prisons,  &c. ;  and  being  adopted  for  these  purposes,  it  is 
gradually  gaining  favour. 

A  great  dissimilarity  exists  in  the  proportion,  form  and  general 
features  of  Egyptian  columns.  In  some  instances,  there  is  no 
uniformity  even  in  those  of  the  same  building,  each  differing 
from  the  others  either  in  its  shaft  or  capital.  For  practical  use 
in  this  country,  Fig.  131  may  be  taken  as  a  standard  of  this 
style.  The  Halls  of  Justice  in  Centre-street,  New- York  city,  is 
a  building  in  general  accordance  with  the  principles  of  Egyptian 
architecture. 

Buildings  in  General. 

218, — That  style  of  architecture  is  to  be  preferred  in  whicn 
utility,  stability  and  regularity,  are  gracefully  blended  with  gran- 
deur and  elegance.  But  as  an  arrangement  designed  for  a  warm 
country  would  be  inappropriate  for  a  colder  climate,  it  would  seem 
that  the  style  of  building  ought  to  be  modified  to  suit  the  wants 
of  the  people  for  whom  it  is  designed.  High  roofs  to  resist  the 
pressure  of  heavy  snows,  and  arrangements  for  artificial  heat,  are 
indispensable  in  northern  climes ;  while  they  would  be  regarded 
as  entirely  out  of  pla<:e  in  buildings  at  the  equator. 

219. — Among  the  Greeks,  architecture  was  employed  chiefly 
upon  their  temples  and  other  large  buildings ;  and  the  proportions 
of  the  orders,  as  determined  by  them,  when  executed  to  such 
large  dimensions,  have  the  happiest  effect.  But  when  used  for 
small  buildings,porticos,  porches,  &c.,  especially  in  country-places, 
they  are  rather  heavy  and  clumsy ;  in  such  cases,  more  slender 
proportions  will  be  found  to  produce  a  better  effect.  The 


AMERICAN    HOUSE-C-i  RPENTER.  10ft 

English  cottage-style  is  rather  more  appropriate,  and  is  berom- 
ing  extensively  practised  for  small  buildings  in  the  country. 

220. — Every  building  should  bear  an  expression  suited  to  its 
destination.  If  it  be  intended  for  national  purposes,  it  should  be 
magnificent — grand ;  for  a  private  residence,  neat  and  modest  ^ 
for  a  banqueting-house,  gay  and  splendid ;  for  a  monument  or 
cemetery,  gloomy — melancholy  ;  or,  if  for  a  church,  majestic  and 
graceful.  By  some  it  has  been  said — "somewhat  dark  and 
gloomy,  as  being  favourable  to  a  devotional  state  of  feeling ;"  but 
such  impressions  can  only  result  from  a  misapprehension  of  the 
nature  of  true  devotion.  "  Her  ways  are  ways  of  pleasantness, 
and  all  her  paths  are  peace."  The  church  should  rather  be  a  type 
of  that  brighter  world  to  which  it  leads. 

221. — However  happily  the  several  parts  of  an  edifice  may  be 
disposed,  and  however  pleasing  it  may  appear  as  a  whole,  yet 
much  depends  upon  its  site,  as  also  upon  the  character  and  style 
of  the  structures  in  its  immediate  vicinity,  and  the  degree  of  cul- 
tivation of  the  adjacent  country.  A  splendid  country-seat  should 
have  the  out-houses  and  fences  in  the  same  style  with  itself,  the 
trees  and  shrubbery  neatly  trimmed,  and  the  grounds  well  cul- 
tivated. 

222. — Europeans  express  surprise  that  so  many  houses  in  this 
country  are  built  of  wood.  And  yet,  in  a  new  country,  where 
wood  is  plenty,  that  this  should  be  so  is  no  cause  for  wonder. 
Still,  the  practice  should  not  be  encouraged.  Buildings  erected 
with  brick  or  stone  are  far  preferable  to  those  of  wood ;  they  are 
more  durable  ;  not  so  liable  to  injury  by  fire,  nor  to  need  repairs ; 
and  will  be  found  in  the  end  quite  as  economical.  A  wooden 
house  is  suitable  for  a  temporary  residence  only ;  and  those  who 
would  bequeath  a  dwelling  to  their  children,  will  endeavour  to 
buil&  with  a  more  durable  material.  Wooden  cornices  and  glit- 
ters, attached  to  brick  houses,  are  objectionable — not  only  on  ac- 
count of  their  frail  nature,  but  also  because  they  render  the  build- 
ing liable  to  destruction  by  fire. 


no 


AMERICAN    HOrSE-CAKPENTER. 


ARCHITECTURE.  Ill 

223. — Dwelling  houses  are  built  of  various  dimensions  and 
styles,  according  to  their  destination  ;  and  to  give  designs  and 
directions  for  their  erection,  it  is  necessary  to  know  their  situa- 
tion and  object.  A  dwelling  intended  for  a  gardener,  would 
require  very  different  dimensions  and  arrangements  from  one 
intended  for  a  retired  gentleman — with  his  servants,  horses, 
&c\ ;  nor  would  a  house  designed  for  the  city  be  appropriate 
for  the  country.  For  city  houses,  arrangements  that  would  be 
convenient  for  one  family  might  be  very  inconvenient  for  twc 
or  more.  Fig.  132,  133,  134,  135,  136,  and  137,  represent  the 
ichnographical  projection,  or  ground-plan,  of  the  floors  of  an 
ordinary  city  house,  designed  to  be  occupied  by  one  family 
only.  Fig.  139  is  an  elevation*  or  front-view,  of  the  same 
house :  all  these  plans  are  drawn  at  the  same  scale — which  is 
that  at  the  bottom  of  Fig.  139. 

Fig.  132  is  a  Plan  of  the  Tinder-Cellar. 

a,  is  the  coal-vault,  6  by  10  feet. 

5,  is  the  furnace  for  heating  the  house. 

c,  d,  are  front  and  rear  areas. 

Fig.  133  is  a  Plan  of  the  Basement. 

a,  is  the  library,  or  ordinary  dining-room,  15  by  20  feet. 

5,  is  the  kitchen,  15  by  22  feet, 
c,  is  the  store-room,  6  by  9  feet. 
<?,  is  the  pantry,  4  by  7  feet. 

«,  is  the  china  closet,  4  by  7  feet. 
/,  is  the  servants'  water-closet. 
g,  is  a  closet. 

h,  is  a  closet  with  a  dumb-waiter  to  the  first  story  abc7«. 
i,  is  an  ash  closet  under  the  front  stoop, 
jf,  is  the  kitchen-range. 

6,  is  the  sink  for  washing  and  drawing  water 
Z,  are  wash  trays. 


112 


AMEKICAN   HOUSE-CARPENTER. 


I 


n«.  IT, 

Second  Story. 


ARCHITECT-TIKE,  113 

Fig.  134  is  a  Plan  of  the  First  Story. 

a,  is  the  parlor,  15  by  34  feet. 

ft,  is  the  dining-room,  16  by  23  feet 

0,  is  the  vestibule. 

«,  is  the  closet  containing  the  dumb-waiter  from  the  basement 

f,  is  the  closet  containing  butler's  sink. 

g,  g,  are  closets. 

A,  is  a  closet  for  hats  and  cloaks. 
i,j,  are  front  and  rear  balconies. 

Fig.  135  is  the  Second  Story. 

a,  a,  are  chambers,  15  by  19  feet. 
ft,  is  a  bed-room,  74  by  13  feet. 

c,  is  the  bath-room,  74  by  13  feet. 

d,  d,  are  dressing-rooms,  6  by  74  feet. 

e,  e,  are  closets. 
ftfi  are  wardrobes. 
<7,  <7,  are  cupboards. 

Fig.  136  is  the  Third  Story. 
a,  a,  are  chambers,  15  by  19  feet. 
ft,  ft,  are  bed-rooms,  74  by  13  feet. 

c,  c,  are  closets. 

d,  is  a  linen  closet,  5  by  7  feet 

e,  e,  are  dressing-closets. 
/,/,  are  wardrobes. 

^,  g,  are  cupboards. 

Fig.  137  is  the  Fourth  Story. 

a,  a,  are  chambers,  14  by  17  feet, 
ft,  ft,  are  bed-rooms,  8J  by  17  feet 

c,  c,  c,  are  closets. 

d,  is  the  step-laddjer  to  the  roof. 

15 


114 


AMERICAN    UOUSE-CAKPENTEK. 


ARCHITECTURE.  115 

Fig,  138  is  the  Section  of  the  House  showing  the  heights  of 

the  several  stories. 

i 

Fig.  139  is  the  Front  Elevation. 

The  size  of  the  house  is  25  feet  front  by  55  feet  deep  ;  this 
is  about  the  average  depth,  although  some  are  extended  to  60 
and  65  feet  in  depth. 

These  are  introduced  to  give  some  general  ideas  of  the  prin- 
ciples to  be  followed  in  designing  city  houses.  In  placing  the 
chimneys  in  the  parlours,  set  the  chimney-breasts  equi-distant 
from  the  ends  of  the  room.  The  basement  chimney-breasts 
may  be  placed  nearly  in  the  middle  of  the  side  of  the  room,  as 
.  there  is  but  one  flue  to  pass  through  the  chimney-breast  above ; 
but  in  the  second  story,  as  there  are  two  flues,  one  from  the 
basement  and  one  from  the  parlour,  the  breast  will  have  to  be 
placed  nearly  perpendicular  over  the  parlour  breast,  so  as  to 
receive  the  flues  within  the  jambs  of  the  fire-place.  As  it  is 
desirable  to  have  the  chimney-breast  as  near  the  middle  of  the 
room  as  possible,  it  may  be  pla'ced  a  few  inches  towards  that 
point  from  over  the  breast  below.  So  in  arranging  those  of 
the  stories  above,  always  make  provision  for  the  flues  from 
below. 

221. — In  placing  the  stairs,  there  should  be  at  least  as  much 
room  in  the  passage  at  the  side  of  the  stairs,  as  upon  them ; 
and  in  regard  to  the  length  of  the  passage  in  the  second  story, 
there  must  be  room  for  the  doors  which  open  from  each  of  the 
principal  rooms  into  the  hall,  and  more  if  the  stairs  require  it. 
Having  assigned  a  position  for  the  stairs  of  the  second  story, 
now  generally  placed  in  the  centre  of  the  depth  of  the  house, 
let  the  winders  of  the  other  stories  be  placed  perpendicularly 
over  and  under  them ;  and  be  careful  to  provide  for  head-- 
room. To  ascertain  this,  when  it  is  doubtful,  it  is  well  to  draw 
a  vertical  section  of  the  whole  stairs ;  but  in  ordinary  cases, 
this  is  not  necessary.  To  dispose  the  windows  properly,  the 


116 


AMERICAN   HOUSE-CARPENTER. 


f_  — r  — f  ."-.'f^-f  '-f  — i  .'••g-TT^t   i-r  -•M 


ARCHITECTURE.  117 

middle  window  of  each  story  should  be  exactly  in  the  middle 
of  the  front ;  but  the  pier  between  the  two  windows  which 
light  the  parlour,  should  be  in  the  centre  of  that  room ;  be- 
cause when  chandeliers  or  any  similar  ornaments,  hang  from 
the  centre-pieces  of  the  parlour  ceilings,  it  is  important,  in 
order  to  give  the  better  effect,  that  the  pier-glasses  at  the 
front  and  rear,  be  in  a  range  with  them.  If  both  these  ob- 
jects cannot  be  attained,  an  approximation  to  each  must  be 
attempted.  The  piers  should  in  no  case  be  less  in  width  than 
the  window  openings,  else  the  blinds  or  shutters  when  thrown 
open  will  interfere  with  one  another ;  in  general  practice,  it  is 
well  to  make  the  outside  piers  f  of  the  width  of  one  of  the 
middle  piers.  When  this  is  desirable,  deduct  the  amount  of 
the  three  openings  from  the  width  of  the  front,  and  the  re- 
mainder will  be  the  amount  of  the  width  of  all  the  piers ; 
divide  this  by  10,  and  the  product  will  be  $  of  a  middle  pier ; 
and  then,  if  the  parlour  arrangements  do  not  interfere,  give 
twice  this  amount  to  each  corner  pier,  and  three  times  the 
same  amount  to  each  of  the  middle  piers. 

PRINCIPLES   OF  ARCHITECTURE. 

225. — In  the  construction  of  the  first  habitations  of  men, 
frail  and  rude  as  they  must  have  been,  the  first  and  principal 
object  was,  doubtless,  utility — a  mere  shelter  from  sun  and 
rain.  But  as  successive  storms  shattered  the  poor  tenement, 
man  was  taught  by  experience  the  necessity  of  building  with 
an  idea  to  durability.  And  when  in  his  walks  abroad,  the 
symmetry,  proportion  and  beauty  of  nature  met  his  admiring 
gaze,  contrasting  so  strangely  with  the  misshapen  and  dispro- 
portioned  work  of  his  own  hands,  he  was  led  to  make  gradual 
changes;  till  his  abode  was  rendered  not  only  commodious 
and  durable,  but  pleasant  in  its  appearance ;  and  building 
became  a  fine  art,  having  utility  for  its  basis. 


118  AMERICAN    HOUSE-CARPENTER. 

226. — 111  all  designs  for  buildings  of  importance,  utility, 
durability  and  beauty,  the  first  great  principles  of  architec- 
ture, should  be  pre-eminent.  In  order  that  the  edifice  be 
useful,  commodious  and  comfortable,  the  arrangement  of  the 
apartments  should  be  such  as  to  fit  them  for  their  several  des- 
tinations ;  for  public  assemblies,  oratory,  state,  visitors,  retir- 
ing, eating,  reading,  sleeping,  bathing,  dressing,  &c. — these 
should  each  have  its  own  peculiar  form  and  situation.  To 
accomplish  this,  and  at  the  same  time  to  make  their  relative 
situation  agreeable  and  pleasant,  producing  regularity  and 
harmony,  require  in  some  instances  much  skill  and  sound 
judgment.  Convenience  and  regularity  are  very  important, 
and  each  should  have  due  attention  ;  yet  when  both  cannot 
be  obtained,  the  latter  should  in  most  cases  give  place  to  the 
former.  A  building  that  is  neither  convenient  nor  regular, 
whatever  other  good  qualities  it  may  possess,  will  be  sure  of 
disapprobation. 

227. — The  utmost  importance  should  be  attached  to  such 
arrangements  as  are  calculated  to  promote  health :  among 
these,  ventilation  is  by  no  means  the  least.  For  this  purpose, 
the  ceilings  of  the  apartments  should  have  a  respectable 
height ;  and  the  sky-light,  or  any  part  of  the  roof  that  can  be 
made  moveable,  should  be  arranged  with  cord  and  pullies,  so 
as  to  be  easily  raised  and  lowered.  Small  openings  near  the 
ceiling,  that  may  be  closed  at  pleasure,  should  be  made  in  the 
partitions  that  separate  the  rooms  from  the  passages — espe- 
cially for  those  rooms  which  are  used  for  sleeping  apartments. 
All  the  apartments  should  be  so  arranged  as  to  secure  their 
being  easily  kept  dry  and  clean.  In  dwellings,  suitable  apart- 
ments should  be  fitted  up  for  lathing  with  all  the  necessary 
apparatus  for  conveying  the  water. 

228. — To  i  sure  stability  in  an  edifice,  it  should  be  designed 
upon  well-known  geometrical  principles :  such  as  science  has  de- 
monstrated to  be  necessary  and  sufficient  fcr  firmness  and  dura 


AMERICAN    HOUSE-CARPENTER.  119 

bility.  It  is  well,  also,  that  it  have  the  appearance  of  stability  as 
well  as  the  reality  ;  for  should  it  seem  tottering  and  unsafe,  the 
sensation  of  fear,  rather  than  those  of  admiration  and  pleasure, 
will  be  excited  in  the  beholder.  To  secure  certainty  and  accu- 
racy in  the  application  of  those  principles,  a  knowledge  of  the 
strength  and  other  properties  of  the  materials  used,  is  indispensa- 
ble ;  and  in  order  that  the  whole  design  be  so  made  as  to  be 
capable  of  execution,  a  practical  knowledge  of  the  requisite 
mechanical  operations  is  quite  important. 

229. — The  elegance  of  an  architectural  design,  although  chiefly 
depending  upon  a  just  proportion  and  harmony  of  the  parts,  will 
be  promoted  by  the  introduction  of  ornaments — provided  this  be 
judiciously  performed.  -For  enrichments  should  not  only  be  of  a 
proper  character  to  suit  the  style  of  the  building,  but  should  also 
have  their  true  position,  and  be  bestowed  in  proper  quantity.  The 
most  common  fault,  and  one  which  is  prominent  in  Roman  archi- 
tecture, is  an  excess  of  enrichment :  an  error  which  is  carefully 
to  be  guarded  against.  But  those  who  take  the  Grecian  models 
for  their  standard,  will  not  be  liable  to  go  to  that  extreme.  In 
ornamenting  a  cornice,  or  any  other  assemblage  of  mouldings,  at 
least  every  alternate  member  shxmld  be  left  plain ;  and  those  that 
are  near  the  eye  should  be  more  finished  than  those  which  are  dis- 
tant. Although  the  characteristics  of  good  architecture  are  utili- 
ty and  elegance,  in  connection  with  durability,  yet  some  buildings 
are  designed  expressly  for  use,  and  others  again  for  ornament :  in 
the  former,  utility,  and  in  the  latter,  beauty,  should  be  the  gov- 
erning principle. 

230. — The  builder  should  be  intimately  acquainted  with  the 
principles  upon  which  the  essential,  elementary  parts  of  a  build- 
ing are  founded.  A  scientific  knowledge  of  these  will  insure 
certainty  and  security,  and  enable  the  mechanic  to  erect  the  most 
extensive  and  lofty  edifices  with  confidence.  The  more  important 
parts  are  the  foundation,  the  column,  the  wall,  the  lintel,  the  arch, 
the  vault,  the  dome  and  the  roof.  A  separate  description  of  the 


120  ARCHITECTURE. 

peculiarities  of  each,  would  seem  to  be  necessary ;  and  cannot 
perhaps  be  better  expressed  than  in  the  following  language  of  a 
modern  writer  on  this  subject. 

231. "In  laying  the  FOUNDATION  of  any  building,  it  is  ne- 
cessary to  dig  to  a  certain  depth  in  the  earth,  to  secure  a  solid 
basis,  below  the  reach  of  frost  and  common  accidents.  The 
most  solid  basis  is  rock,  or  gravel  which  has  not  been  moved. 
Next  to  these  are  clay  and  sand,  provided  no  other  excavations 
have  been  made  in  the  immediate  neighbourhood.  From  this 
basis  a  stone  wall  is  carried  up  to  the  surface  of  the  ground,  and 
constitutes  the  foundation.  Where  it  is  intended  that  the  super- 
structure shall  press  unequally,  as  at  its  piers,  chimneys,  or 
columns,  it  is  sometimes  of  use  to  occupy  the  space  between  the 
points  of  pressure  by  an  inverted  arch.  This  distributes  the 
pressure  equally,  and  prevents  the  foundation  from  springing  be- 
tween the  different  points.  In  loose  or  muddy  situations,  it  is 
always  unsafe  to  build,  unless  we  can  .reach  the  solid  bottom 
below.  In  salt  marshes  and  flats,  this  is  done  by  depositing  tim- 
bers, or  driving  wooden  piles  into  the  earth,  and  raising  walls 
upon  them.  The  preservative  quality  of  the  salt  will  keep  these 
timbers  unimpaired  for  a  great  length  of  time,  and  makes  the 
foundation  equally  secure  with  one  of  brick  or  stone. 

232. — The  simplest  member  in  any  building,  though  by  no 
means  an  essential  one  to  all,  is  the  COLUMN,  or  pillar.  This  is 
a  perpendicular  part,  commonly  of  equal  breadth  and  thickness, 
not  intended  for  the  purpose  of  enclosure,  but  simply  for  the  sup- 
port of  some  part  of  the  superstructure.  The  principal  force 
which  a  column  has  to  resist,  is  that  of  perpendicular  pressure. 
In  its  shape,  the  shaft  of  a  column  should  not  be  exactly  cylin- 
drical, but,  since  the  lower  part  must  support  the  weight  of  the 
superior  part,  in  addition  to  the  weight  which  presses  equally  on 
the  whole  column,  the  thickness  should  gradually  decrease  from 
bottom  to  top.  The  outline  of  columns  should  be  a  little  curved, 
•o  as  to  represent  a  portion  of  a  very  long  spheroid,  or  paraboloid, 


AMERICAN    HOUSE-CARPENTER.  121 

rather  than  of  a  cone.  This  figure  is  the  joint  result  of  two  cal- 
culations, independent  of  beauty  of  appearance.  One  of  these 
is,  that  the  form  best  adapted  for  stability  of  base  is  that  of  a 
cone;  the  other  is,  that  the  figure,  which  would  be  of  equal 
strength  throughout  for  supporting  a  superincumbent  weight, 
would  be  generated  by  the  revolution  of  two  parabolas  round  the 
axis  of  the  column,  the  vertices  of  the  curves  being  at  its  ex- 
tremities. The  swell  of  the  shafts  of  columns  was  called  the  en- 
tasis by  the  ancients.  It  has  been  lately  found,  that  the  columns 
of  the  Parthenon,  at  Athens,  which  have  been  commonly  sup- 
posed straight,  deviate  about  an  inch  from  a  straight  line,  and 
that  their  greatest  swell  is  at  about  one  third  of  their  height. 
Columns  in  the  antique  orders  are  usually  made  to  diminish  one 
sixth  or  one  seventh  of  their  diameter,  and  sometimes  even  one 
fourth.  The  Gothic  pillar  is  commonly  of  equal  thickness 
throughout. 

233. — The  WALL,  another  elementary  part  of  a  building,  may 
be  considered  as  the  lateral  continuation  of  the  column,  answer- 
ing the  purpose  both  of  enclosure  and  support.  A  wall  must 
diminish  as  it  rises,  for  the  same  reasons,  and  in  the  same  propor- 
tion, as  the  column.  It  must  diminish  still  more  rapidly  if  it  ex- 
tends through  several  stories,  supporting  weights  at  different 
heights.  A  wall,  to  possess  the  greatest  strength,  must  also  con- 
sist of  pieces,  the  upper  and  lower  surfaces  of  which  are  horizon- 
tal arid  regular,  not  rounded  nor  oblique.  The  walls  of  most  of 
the  ancient  structures  which  have  stood  to  the  present  time,  are 
constructed  in  this  manner,  and  frequently  have  their  stones  bound 
together  with  bolts  and  cramps  of  iron.  The  same  method  is 
adopted  in  such  modern  structures  as  are  intended  to  possess  great 
strength  and  durability,  and,  in  some  cases,  the  stones  are  even 
clove-tailed  together,  as  in  the  light-houses  at  Eddystone  and  Bell 
Rock,  But  many  of  our  modern  stone  walls,  for  the  sake  ol 
cheapness,  have  only  one  face  of  the  stones  squared,  the  innei 
half  of  the  wall  being  completed  with  brick ;  so  that  they  can, 
16 


122  ARCHITECTURE. 

in  reality,  be  considered  only  as  brick  walls  faced  with  stone 
Such  walls  are  said  to  be  liable  to  become  convex  outwardly,  from 
the  difference  in  the  shrinking  of  the  cement.  Rubble  walls  are 
made  of  rough,  irregular  stones,  laid  in  mortar.  The  stones 
should  be  broken,  if  possible,  so  as  to  produce  horizontal  surfaces 
The  coffer  walls  of  the  ancient  Romans  were  made  by  enclosing 
successive  portions  of  the  intended  wall  in  a  box,  and  filling  it 
with  stones,  sand,  and  mortar,  promiscuously.  This  kind  of 
structure  must  have  been  extremely  insecure.  The  Pantheon, 
and  various  other  Roman  buildings,  are  surrounded  with  a  double 
brick  wall,  having  its  vacancy  filled  up  with  loose  bricks  and 
cement.  The  whole  has  gradually  consolidated  into  a  mass  ot 
great  firmness. 

The  reticulated  walls  of  the  Romans,  having  bricks  with 
oblique  surfaces,  would,  at  the  present  day,  be  thought  highly 
unphilosophical.  Indeed,  they  could  not  long  have  stood,  had  it 
not  been  for  the  great  strength  of  their  cement.  Modern  brick 
walls  are  laid  with  great  precision,  and  depend  for  firmness  more 
upon  their  position  than  upon  the  strength  of  their  cement.  The 
.bricks  being  laid  in  horizontal  courses,  and  continually  overlaying 
each  other,  or  breaking  joints,  the  whole  mass  is  strongly  inter- 
woven, and  bound  together.  Wooden  walls,  composed  of  timbers 
covered  with  boards,  are  a  common,  but  more  perishable  kind. 
They  require  to  be  constantly  covered  with  a  coating  of  a  foreign 
substance,  as  paint  or  plaster,  to  preserve  them  from  spontaneous 
decomposition.  In  some  parts  of  France,  and  elsewhere,  a  kind 
of  wall  is  made  of  earth,  rendered  compact  by  ramming  it  in 
moulds  or  cases.  This  method  is  called  building  in  pise,  and  is 
much  more  durable  than  the  nature  of  the  material  would  lead 
us  to  suppose.  Walls  of  all  kinds  are  greatly  strengthened  by 
angles  and  curves,  also  by  projections,  such  as  pilasters,  chimneys 
and  buttresses.  These  projections  serve  to  increase  the  breadth 
of  the  foundation,  and  are  always  to  be  made  use  of  in  large 
buildings,  and  in  walls  of  considerable  length. 


AMERICAN    HOUSE-CARPENTER.  123 

234.— The  LINTEL,  or  beam,  extends  in  a  right  line  over  a 
Vacant  space,  from  one  column  or  wall  to  another.  The  strength 
of  the  lintel  will  be  greater  in  proportion  as  its  transverse  vertical 
diameter  exceeds  the  horizontal,  the  strength  being  always  as  the 
square  of  the  depth.  The  floor  is  the  lateral  continuation  or 
connection  of  beams  by  means  of  a  covering  of  boards. 

235. — The  ARCH  is  a  transverse  member  of  a  building,  an- 
swering the  same  purpose  as  the  lintel,  but  vastly  exceeding  it  in 
strength.  The  arch,  unlike  the  lintel,  may  consist  of  any  num- 
ber of  constituent  pieces,  without  impairing  its  strength.  It  is, 
however,  necessary  that  all  the  pieces  should  possess  a  uniform 
shape, — the  shape  of  a  portion  of  a  wedge, — and  that  the  joints, 
formed  by  the  contact  of  their  surfaces,  should  point  towards  a 
common  centre.  In  this  case,  no  one  portion  of  the  arch  can  be 
displaced  or  forced  inward ;  and  the  arch  cannot  be  broken  by 
any  force  which  is  not  sufficient  to  crush  the  materials  of  which 
it  is  made.  In  arches  made  of  common  bricks,  the  sides  of  which 
are  parallel,  any  one  of  the  bricks  might  be  forced  inward,  were 
it  not  for  the  adhesion  of  the  cement.  Any  two  of  the  bricks, 
however,  by  the  disposition  of  their  mortar,  cannot  co^ective- 
ly  be  forced  inward.  An  arch  of  the  proper  form,  when  com- 
plete, is  rendered  stronger,  instead  of  weaker,  by  the  pressure  of 
a  considerable  weight,  provided  this  pressure  be  uniform.  While 
building,  however,  it  requires  to  be  supported  by  a  centring  of 
the  shape  of  its  internal  surface,  until  it  is  complete.  The  upper 
stone  of  an  arch  is  called  the  key-stone,  but  is  not  more  essential 
than  any  other.  In  regard  to  the  shape  of  the  arch,  its  most 
simple  form  is  that  of  the  semi-circle.  It  is,  however,  very  fre- 
quently a  smaller  arc  of  a  circle,  and,  still  more  frequently,  a  por- 
tion of  an  ellipse.  The  simplest  theory  of  an  arch  supporting 
itself  only,  is  that  of  Dr.  Hooke.  The  arch,  when  it  has  only 
its  own  weight  to  bear,  may  be  considered  as  the  inversion  of  a 
chain,  suspended  at  each  end.  The  chain  hangs  in  such  a  form, 
that  the  weight  of  each  link  or  portion  is  held  in  equilibrium  by 


124  ARCHITECTURE. 

the  result  of  two  forces  acting  at  its  extremities ;  and  these  forces, 
or  tensions,  are  produced,  the  one  by  the  weight  of  the  portion  of 
the  chain  below  the  link,  the  other  by  the  same  weight  increased 
by  that  of  the  link  itself,  both  of  them  acting  originally  in  a  ver- 
tical direction.  Now,  supposing  the  chain  inverted,  so  as  to  con- 
stitute an  arch  of  the  same  form  and  weight,  the  relative  situa- 
tions of  the  forces  will  be  the  same,  only  they  will  act  in  contrary 
directions,  so  that  they  are  compounded  in  a  similar  manner,  and 
balance  each  other  on  the  same  conditions. 

The  arch  thus  formed  is  denominated  a  catenary  arch.  In 
common  cases,  it  differs  but  little  from  a  circular  arch  of  the  extent 
of  about  one  third  of  a  whole  circle,  and  rising  from  the  abut- 
ments with  an  obliquity  of  about  30  degrees  from  a  perpendicu- 
lar. But  though  the  catenary  arch  is  the  best  form  for  support- 
ing its  own  weight,  and  also  all  additional  weight  which  presses 
in  a  vertical  direction,  it  is  not  the  best  form  to  resist  lateral 
pressure,  or  pressure  like  that  of  fluids,  acting  equally  in  all  direc- 
tions. Thus  the  arches  of  bridges  and  similar  structures,  when 
covered  with  loose  stones  and  earth,  are  pressed  sideways,  as  well 
as  vertically,  in  the  same  manner  as  if  they  supported  a  weight 
of  fluid.  In  this  case,  it  is  necessary  that  the  arch  should  arise 
more  perpendicularly  from  the  abutment,  and  that  its  general 
figure  should  be  that  of  the  longitudinal  segment  of  an  ellipse. 
In  small  arches,  in  common  buildings,  where  the  disturbing 
force  is  not  great,  it  is  of  little  consequence  what  is  the  shape  ot 
the  curve.  The  outlines  may  even  be  perfectly  straight,  as  in  the 
tier  of  bricks  which  we  frequently  see  over  a  window.  This  is, 
strictly  speaking,  a  real  arch,  provided  the  surfaces  of  the  bricks 
tend  towards  a  common  centre.  It  is  the  weakest  kind  of  arch, 
and  a  part  of  it  is  necessarily  superfluous,  since  no  greater  portion 
can  act  in  supporting  a  weight  above  it,  than  can  be  included  be- 
tween two  curved  or  arched  lines. 

Besides  the  arches  already  mentioned,  various  others  are  in  use 
The  acute  or  lancet  arch,  much  used  in  Gothic  architecture,  is 


AMERICAN    HOUSE-CARPENTER.  125 

described  usually  from  two  centres  outside  the  arch.  It  is  a 
strong  arch  for  supporting  vertical  pressure.  The  rampant  arch 
is  one  in  which  the  two  ends  spring  from  unequal  heights.  The 
horse-shoe  or  Moorish  arch  is  described  from  one  or  more  centres 
placed  above  the  base  line.  In  this  arch,  the  lower  parts  are  in 
danger  of  being  forced  inward.  The  ogee  arch  is  concavo-con- 
vex, and  therefore  fit  only  for  ornament.  In  describing  archeSj 
the  upper  surface  is  called  the  extrados,  and  the  inner,  the  in- 
trados.  The  springing  lines  are  those  where  the  intrados  meets 
the  abutments,  or  supporting  walls.  The  span  is  the  distance 
from  one  springing  line  to  the  other.  The  wedge-shaped  stones, 
which  form  an  arch,  are  sometimes  called  voussoirs,  the  upper- 
most being  the  key-stone.  The  part  of  a  pier  from  which  an 
arch  springs  is  called  the  impost,  and  the  curve  formed  by  the 
upper  side  of  the  voussoirs,  the  archivolt.  It  is  necessary  that 
the  walls,  abutments  and  piers,  on  which  arches  are  supported, 
should  be  so  firm  as  to  resist  the  lateral  thrust,  as  well  as  vertical 
pressure,  of  the  arch.  It  will  at  once  be  seen,  that  the  lateral  or 
sideway  pressure  of  an  arch  is  very  considerable,  when  we  recol- 
lect that  every  stone,  or  portion  of  the  arch,  is  a  wedge,  a  part  of 
whose  force  acts  to  separate  the  abutments.  For  want  of  atten- 
tion to  this  circumstance,  important  mistakes  have  been  committed, 
the  strength  of  buildings  materially  impaired,  and  their  ruin  ac- 
celerated. In  some  cases,* the  want  of  lateral  firmness  in  the 
walls  is  compensated  by  a  bar  of  iron  stretched  across  the  span  ot 
the  arch,  and  connecting  the  abutments,  like  the  tie-beam  of  a 
roof.  This  is  the  case  in  the  cathedral  of  Milan  and  some  other 
Gothic  buildings. 

In  an  arcade,  or  continuation  of  arches,  it  is  only  necessary  that 
the  outer  supports  of  the  terminal  arches  should  be  strong  enough 
to  resist  horizontal  pressure.  In  the  intermediate  arches,  the  lat- 
eral force  of  each  arch  is  counteracted  by  the  opposing  lateral 
force  of  the  one  contiguous  to  it.  In  bridges,  however,  where 
individual  arches  are  liable  to  be  destroyed  by  accident,  it  is  desi 


126  ARCHITECTURE. 

rable  that  each  of  the  piers  should  possess  sufficient  horizontal 
strength  to  resist  the  lateral  pressure  of  the  adjoining  arches. 

236. The  VAULT  is  the  lateral  continuation  of  an  arch,  serving 

to  cover  an  area  or  passage,  and  bearing  the  same  relation  to  the 
arch  that  the  wall  does  to  the  column.  A  simple  vault  is  con- 
structed on  the  principles  of  the  arch,  and  distributes  its  pressure 
equally  along  the  walls  or  abutments.  A  complex  or  groined 
vault  is  made  by  two  vaults  intersecting  each  other,  in  which 
case  the  pressure  is  thrown  upon  springing  points,  and  is  greatly 
increased  at  those  points.  The  groined  vault  is  common  in 
Gothic  architecture. 

237. — The  DOME,  sometimes  called  cupola,  is  a  concave  cover- 
ing to  a  building,  or  part  of  it,  and  may  be  either  a  segment  of  a 
sphere,  of  a  spheroid,  or  of  any  similar  figure.  When  built  of 
stone,  it  is  a  very  strong  kind  of  structure,  even  more  so  than  the 
arch,  since  the  tendency  of  each  part  to  fall  is  counteracted,  not 
only  by  those  above  and  below  it,  but  also  by  those  on  each  side. 
It  is  only  necessary  that  the  constituent  pieces  should  have  a 
common  form,  and  that  this  form  should  be  somewhat  like  the 
frustum  of  a  pyramid,  so  that,  when  placed  in  its  situation,  its 
four  angles  may  point  toward  the  centre,  or  axis,  of  the  dome. 
During  the  erection  of  a  dome,  it  is  not  necessary  that  it  should 
be  supported  by  a  centring,  until  complete,  as  is  done  in  the  arch. 
Each  circle  of  stones,  when  laid,  is  Capable  of  supporting  itself 
without  aid  from  those  above  it.  It  follows  that  the  dome  may 
be  left  open  at  top,  without  a  key-stone,  and  yet  be  perfectly 
secure  in  this  respect,  being  the  reverse  of  the  arch.  The  dome 
of  the  Pantheon,  at  Rome,  has  been  always  open  at  top,  and  yet 
has  stood  unimpaired  for  nearly  2000  years.  The  upper  circle 
of  stones,  though  apparently  the  weakest,  is  nevertheless  often 
made  to  support  the  additional  weight  of  a  lantern  or  tower  above 
it.  In  several  of  the  largest  cathedrals,  there  are  two  domes,  one 
within,  the  other,  which  contribute  their  joint  support  to  the  lan- 
tern, which  rests  upon  the  top.  In  these,  buildings,  the  dome 


AMERICAN    HOUSE-CARPENTER. 


127 


rests  upon  a  circular  wall,  which  is  supported,  in  its  turn,  by 
arches  upon  massive  pillars  or  piers.  This  construction  is  called 
building  upon  pendentives,  and  gives  open  space  and  room  for 
passage  beneath  the  dome.  The  remarks  which  have  been  made 
in  regard  to  the  abutments  of  the  arch,  apply  equally  to  the  walls 
immediately  supporting  a  dome.  They  must  be  of  sufficient 
thickness  and  solidity  to  resist  the  lateral  pressure  of  the  dome, 
which  is  very  great.  The  walls  of  the  Roman  Pantheon  are  of 
great  depth  and  solidity.  In  order  that  a  dome  in  itself  should  be 
perfectly  secure,  its  lower  parts  must  not  be  too  nearly  vertical, 
since,  in  this  case,  they  partake  of  the  nature  of  perpendicular 
walls,  and  are  acted  upon  by  the  spreading  force  of  the  parts  above 
them.  The  dome  of  St.  Paul's  church,  in  London,  and  some 
others  of  similar  construction,  are  bound  with  chains  or  hoops  ot 
iron,  to  prevent  them  from  spreading  at  bottom.  Domes  which 
are  made  of  wood  depend,  in  part,  for  their  strength,  on  their  in- 
ternal carpentry.  The  Halle  du  Bled,  in  Paris,  had  originally  a 
wooden  dome  more  than  200  feet  in  diameter,  and  only  one  foot 
in  thickness.  This  has  since  been  replaced  by  a  dome  of  iion 
(See  Art.  389.) 

238. — The  ROOF  is  the  most  common  and  cheap  method  of 
covering  buildings,  to  protect  them  from  rain  and  other  effects  of 
the  weather.  It  is  sometimes  flat,  but  more  frequently  oblique,  in 
its  shape.  The  flat  or  platform-roof  is  the  least  advantageous  for 
shedding  rain,  and  is  seldom  used  in  northern  countries.  The 
pent  roof,  consisting  of  two  oblique  sides  meeting  at  top,  is  the 
most  common  form.  These  roofs  are  made  steepest  in  cold  cli- 
mates, where  they  are  liable  to  be  loaded  with  snow.  Where  the 
four  sides  of  the  roof  are  all  oblique,  it  is  denominated  a  hipped 
roof,  and  where  there  are  two  portions  to  the  roof,  of  different  ob- 
liquity, it  is  a  curb,  or  mansard  roof.  In  modern  times,  roofs 
are  made  almost  exclusively  of  wood,  though  frequently  covered 
with  incombustible  materials.  The  internal  structure  or  carpen- 
try of  roofs  is  a  subject  of  considerable  mechanical  contrivance, 


128  ARCHITECTURE. 

The  roof  is  supported  by  rafters,  which  abut  on  the  walls  on 
each  side,  like  the  extremities  of  an  arch.  If  no  other  timbers 
existed,  except  the  rafters,  they  would  exert  a  strong  lateral  pres- 
sure on  the  walls,  tending  to  separate  and  overthrow  them.  To 
counteract  this  lateral  force,  a  tie-beam,  as  it  is  called,  extends 
across,  receiving  the  ends  of  the  rafters,  and  protecting  the  wall 
from  their  horizontal  thrust.  To  prevent  the  tie-beam  from 
sagging,  or  bending  downward  with  its  own  weight,  a  king- 
post is  erected  from  this  beam,  to  the  upper  angle  of  the  rafters, 
serving  to  connect  the  whole,  and  to  suspend  the  weight  of  the 
beam.  This  is  called  trussing.  Queen-posts  are  sometimes 
added,  parallel  to  the  king-post,  in  large  roofs  ;  also  various  other 
connecting  timbers.  In  Gothic  buildings,  where  the  vaults  do 
not  admit  of  the  use  of  a  tie-beam,  the  rafters  are  prevented  from 
spreading,  as  in  an  arch,  by  the  strength  of  the  buttresses. 

In  comparing  the  lateral  pressure  of  a  high  roof  with  that  of  a 
low  one,  the  length  of  the  tie-beam  being  the  same,  it  will  be 
seen  that  a  high  roof,  from  its  containing  most  materials,  may 
produce  the  greatest  pressure,  as  far  as  weight  is  concerned.  On 
the  other  hand,  if  the  weight  of  both  be  equal,  then  the  low  roof 
will  exert  the  greater  pressure  ;  and  this  will  increase  in  propor- 
tion to  the  distance  of  the  point  at  which  perpendiculars,  drawn 
from  the  end  of  each  rafter,  would  meet.  In  roofs,  as  well  as  in 
wooden  domes  and  bridges,  the  materials  are  subjected  to  an  in- 
ternal strain,  to  resist  which,  the  cohesive  strength  of  the  material 
is  relied  on.  On  this  account,  beams  should,  when  possible,  be 
of  one  piece.  Where  this  cannot  be  effected,  two  or  more  beams 
are  connected  together  by  splicing:  Spliced  beams  are  never  so 
strong  as  whole  ones,  yet  they  may  be  made  to  approach  the  same 
strength,  by  affixing  lateral  pieces,  or  by  making  the  ends  overlay 
each  other,  and  connecting  them  with  bolts  and  straps  of  iron. 
The  tendency  to  separate  is  also  resisted,  by  letting  the  two  pieces 
into  each  Kher  by  the  process  called  scarfing.  Mortices,  in- 


AMERICAN    HOUSE-CARPENTER.  129 

tended  to  truss  or  suspend  one  piece  by  another,  should  be  formed 
upon  similar  principles. 

Roofs  in  the  United  States,  after  being  boarded,  receive  a  se- 
condary covering  of  shingles.  When  intended  to  be  incombustible, 
they  are  covered  with  slates  or  earth ern  tiles,  or  with  sheets  of  lead, 
copper  or  tinned  iron.  Slates  are  preferable  to  tiles,  being  lighter, 
and  absorbing  less  moisture.  Metallic  sheets  are  chiefly  used  for 
flat  roofs,  wooden  domes,  and  curved  and  angular  surfaces,  which 
require  a  flexible  material  to  cover  them,  or  have  not  a  sufficient 
pitch  to  shed  the  rain  from  slates  or  shingles.  Various  artificial 
compositions  are  occasionally  used  to  cover  roofs,  the  most  com- 
mon of  which  are  mixtures  of  tar  with  lime,  and  sometimes  with 
sand  and  gravel." — Ency,  Am.  (See  Art.  354.) 

17 


SECTION  III,— MOULDINGS,  CORNICES, 


MOULDINGS. 

239. — A  moulding  is  so  called,  because  of  its  being  of  the 
same  determinate  shape  along  its  whole  length,  as  though  the 
whole  of  it  had  been  cast  in  the  same  mould  or  form.  The  regular 
mouldings,  as  found  in  remains  of  ancient  architecture,  are  eight 
in  number  ;  and  are  known  by  the  following  names  : 


Annulet,  band,  cincture,  fillet,  listel  or  square. 


)  Astragal  or  bead. 


Torus  01  tore. 

Ki«.  142. 


t'ig.  143. 


Scotia,  trochilus  or  mouth, 


^         Ovolo,  quarter-round  or  echinus. 

Fig.  144 


AMERICAN    HOUSE-CARPENTER. 


131 


fig.  145. 


Cavetto,  cove  or  hollow. 


Fig.  147. 


Cymatium,  or  cyma-recta. 


Inverted  cymatium,  or  cyma-reversa 


Ogee. 


Some  of  the  terms  are  derived  thus :  fillet,  from  the  French 
worcl^-,  thread.  Astragal,  from  astragalos,  a  bone  of  the  heel 
— or  the  curvature  of  the  heel.  Bead,  because  this  moulding, 
when  properly  carved,  resembles  a  string  of  beads.  Torus,  or 
tore,  the  Greek  for  rope,  which  it  resembles,  when  on  the  base  of 
a  column.  Scotia,  from  shotia,  darkness,  because  of  the  strong 
shadow  which  its  depth  produces,  and  which  is  increased  by  the 
projection  of  the  torus  above  it.  Ovolo,  from  ovum,  an  egg, 
which  this  member  resembles,  when  carved,  as  in  the  Ionic  capi- 
tal. Cavetto,  from  cavus,  hollow.  Cymatium,  from  kumaton 
a  wave. 

240. — Neither  of  these  mouldings  is  peculiar  to  any  one  of  the 
orders  of  architecture,  but  each  one  is  common  to  all ;  and  al- 
though each  has  its  appropriate  use,  yet  it  is  by  no  means  con- 
fined to  any  certain  position  in  an  assemblage  of  mouldings 
The  use  of  the  fillet  is  to  bind  the  parts,  as  also  that  of  the  astra- 
gal and  torus,  which  resemble  ropes.  The  ovolo  and  cyma-re- 
versa are  strong  at  their  upper  extremities,  and  are  therefore  used 
to  support  projecting  parts  above  them.  The  cyma-recta  and 
cavetto,  being  weak  at  their  upper  extremities,  are  not  used  as 
supporters,  but  are  placed  uppermost  to  cover  and  shelter  the 
other  parts.  The  scotia  is  introduced  in  the  base  of  a  column,  to 


132  MOULDINGS.  CORNICES,  &C. 

separate  the  upper  and  lower  torus,  and  to  produce  a  pleasing 
variety  and  relief.  The  form  of  the  bead,  and  that  of  the  torus, 
is  the  same ;  the  reasons  for  giving  distinct  names  to  them  are, 
that  the  torus,  in  every  order,  is  always  considerably  larger  than 
Ihe  bead,  and  is  placed  among  the  base  mouldings,  whereas  the 
bead  is  never  placed  there,  but  on  the  capital  or  entablature ;  the 
torus,  also,  is  seldom  carved,  whereas  the  bead  is;  and  while  the 
torus  among  the  Greeks  is  frequently  elliptical  in  its  form,  the 
bead  retains  its  circular  shape.  While  the  scotia  is  the  reverse  of 
the  torus,  the  cavetto  is  the  reverse  of  the  ovolo,  and  the  cyma- 
recta  and  cyma-re versa  are  combinations  of  the  ovolo  and  cavetto. 

241. — The  curves  of  mouldings,  in  Roman  architecture,  were 
most  generally  composed  of  parts  of  circles ;  while  those  of  the 
Greeks  were  almost  always  elliptical,  or  of  some  one  of  the  conic 
sections,  but  rarely  circular,  except  in  the  case  of  the  bead,  which 
was  always,  among  both  Greeks  and  Romans,  of  the  form  of  a 
semi-circle.  Sections  of  the  cone  afford  a  greater  variety  ol 
forms  than  those  of  the  sphere ;  and  perhaps  this  is  one  reason 
why  the  Grecian  architecture  so  much  excels  the  Roman.  The 
quick  turnings  of  the  ovolo  and  cyma-reversa,  in  particular,  when 
exposed  to  a  bright  sun,  cause  those  narrow,  well-defined  streaks 
of  light,  which  give  life  and  splendour  to  the  whole. 

242. — A  profile  is  an  assemblage  of  essential  parts  and  mould- 
ings. That  profile  produces  the  happiest  effect  which  is  com- 
posed of  but  few  members,  varied  in  form  and  size,  and  arranged 
so  that  the  -plane  and  the  curved  surfaces  succeed  each  other  al- 
ternately. 

243. —  To  describe  the  Grecian  torus  and  scotia.  Join  the 
extremities,  a  and  6,  (Fig.  148;)  and  from/,  the  given  projection 
ot  the  moulding,  draw/o,  at  right  angles  to  the  fillets;  from  6, 
draw  b  h,  at  right  angles  to  a  b  ;  bisect  a  b  in  c  ;  join  /  ^.d  c, 
and  upon  c,  with  the  radius,  c/,  describe  the  arc,  /  h,  cutting  b  h 
in  h  ;  through  c,  draw  d  e,  parallel  with  the  fillets  ;  make  d  c  and 
c  e,  each  equal  to  b  h  ;  then  d  e  and  a  b  will  be  conjugate  diame- 


AMERICAN    HOUSE-CARPENTER. 


133 


Fig.  148. 


ters  of  the  required  ellipse.  To  describe  the  curve  by  intersec- 
tion of  lines,  proceed  as  directed  at  Art.  118  and  note ;  by  a 
trammel,  see  Art.  116  ;  and  to  find  the  foci,  in  order  to  describe  it 
with  a  string,  see  Art.  115. 


Fig.  149. 


Fig.  150. 


244. — Fig.  149  to  156  exhibit  various  modifications  of  the 
Grecian  ovolo,  sometimes  called  echinus.     Fig.  149  to  153  ar« 


134 


MOULDINGS,    CORNICES,    &C. 


Fig.  151. 


Fig.  152. 


1'ig.  156. 


elliptical,  a  b  and  b  c  being  given  tangents  to  the  curve;  parallel 
to  which,  the  semi-conjugate  diameters,  a  d  and  d  c,  are  drawn. 
In  Fig.  149  and  150,  the  lines,  a  d  and  d  c,  are  semi-axes,  the 
tangents,  a  b  and  b  c,  being  at  right  angles  to  each  other.  To 
draw  the  curve,  see  Art.  118.  In  Fig.  153,  the  curve  is  para- 
bolical, and  is  drawn  according  to  Art.  127.  In  Fig.  155  and  156, 
the  curve  is  hyperbolical,  being  described  according  to  Art.  128. 
The  length  of  the  transverse  axis,  a  b,  being  taken  at  pleasure 
in  order  to  flatten  the  curve,  a  b  should  be  made  short  in  propor- 
tion to  a  c. 


AMERICAN    HOUSE-CARPENTER. 


135 


Fig.    53 


Fig.  157. 


245.—  To  describe  the  Grecian  cavetto,  (Fig.  157  and  158,) 
having  the  height  and  projection  given,  see  Art.  118. 


a 

! 

I 

1 

J 

d 

60. 

Fig.l 

Fig.  159. 

246. —  To  describe  the  Grecian  cyma-recta.  When  the  pro- 
jection is  more  than  the  height,  as  at  Fig.  150,  make  a  b  equal 
to  the  .height,  and  divide  q  b  c  d  into  4  equal  parallelograms ; 
then  proceed  as  directed  in  note  to  Art.  118.  When  the  projec- 
tion is  less  than  the  height,  draw  d  a,  (Fig.  160,)  at  right  angles 
to  a  b  ;  complete  tire  rectangle,  abed;  divide  this  into  4  equal 
rectangles,  and  proceed  according  to  Art.  118. 


Fig.  161. 


247. —  To  describe  the  Grecian  cyma-reversa.      When  the 


136 


MOULDINGS,    CORNICES,    &C. 


projection  is  more  than  the  height,  as  at  Fig.  161,  proceed  as  di 
reeled  for  the  last  figure ;  the  curve  being  the  same  as  that,  the 
position  only  being  changed.  When  the  projection  is  less  than 
the  height,  draw  a  d,  (Fig.  162,)  at  right  angles  to  the  fillet 
make  a  d  equal  to  the  projection  of  the  moulding  :  then  proceed 
as  directed  for  Fig.  159. 

248. — Roman  mouldings  are  composed  of  parts  of  circles,  and 
have,  therefore,  less  beauty  of  form  than  the  Grecian.  The  bead 
and  torus  are  of  the  form  of  the  semi-circle,  and  the  scotia,  also, 
in  some  instances ;  but  the  latter  is  often  composed  of  two  quad- 
rants, having  different  radii,  as  at  Fig.  163  and  164,  which  re- 
semble the  elliptical  curve.  The  ovolo  and  cavetto  are  generally 
a  quadrant,  but  often  less.  When  they  are  less,  as  at  Fig.  167, 
the  centre  is  found  thus  :  join  the  extremities,  a  and  6,  and  bisect 
a  b  in  c  ;  from  e,  and  at  right  angles  to  a  b,  draw  c  d,  cutting  a 
level  line  drawn  from  a  in  d  ;  then  d  will  be  the  centre.  This 
moulding  projects  less  than  its  height.  When  the  projection  is 
more  than  the  height,  as  at  Fig.  169,  extend  the  line  from  c  until 


Fig.  164. 


Fig.  165. 


Fig.  1G6. 


AMERICAN    HOUSE-CARPENTER. 


1ST 


Fig.  16T.  , 


16S. 


Fig.  169. 


Fig.  1TO. 


Fig.  171. 


Fig.  172. 


Fig.  173. 


JS 


Fig.  174. 


133 


MOULDINGS,    CORMCES,    &C 


Fig  175. 


Fig.  176. 


Fig.  177. 


it  cuts  a  perpendicular  drawn  from  <z,  as  at  d  /  and  that  will 
be  the  centre  of  the  curve.  In  a  similar  manner,  the  centres 
are  found  for  the  mouldings  at  Fig.  164,  168,  170,  173,  174, 
175,  and  176.  The  centres  for  the  curves  at  Fig.  177  and  178, 
are  found  thus  :  bisect  the  line,  d  5,  at  c  /  upon  «,  c  and  5,  suc- 
cessively, with  a  c  or  c  I  for  radius,  describe  arcs  intersecting 
at  d  and  d  /  then  those  intersections  will  be  the  centres. 

249. — Fig.  179  to  186  represent  mouldings  of  modern  inven- 
tion. They  have  been  quite  extensively  and  successfully  used 
in  inside  finishing.  Fig.  179  is  appropriate  for  a  bed-moulding 
under  a  low  projecting  shelf,  and  is  frequently  used  under  man- 
tle-shelves. The  tangent,  i  A,  is  found  thus :  bisect  the  lino,  a  I, 
at  c,  and  b  c  at  dj  from  d,  draw  d  e,  at  right  angles  to  e  b  /  from 
&,  draw  bf,  parallel  to  e  d  j  upon  Z>,  with  1)  d  for  radius,  describe 
the  arc,  df  ;  divide  this  arc  into  7  equal  parts,  and  set  one  of  the 
parts  from  s,  the  limit  of  the  projection,  to  o  /  make  o  h  equal  to 
v  e  ;  from  A,  through  c,  draw  the  tangent,  h  i  /  divide  5  A,  /*  c,  c  i 
and  :  a,  each  into  a  like  number  of  equal  parts,,  and  draw  the  in- 


AMERICAN    HOUSE-CARPENTER, 


139 


rig.  isi. 


* 

140 


MOULDINGS,    CORNICES,    &C. 


Fig.  182. 


Fig.  183. 


Fig   184. 


Fig.  185. 


Fig.  186. 


tersecting  lines  as  directed  at  Art.  89.  If  a 'bolder  form  is 
desired,  draw  the  tangent,  i  A,  nearer  horizontal,  and  describe 
an  elliptic  curve  as  shown  in  Fig.  148  and  181.  Fig.  180  is 
much  used  on  base,  or  skirting  of  rooms,  and  in  deep  panelling. 
The  curve  is  found  in  the  same  manner  as  that  of  Fig.  If  9.  In 
this  case,  however,  where  the  moulding  has  so  little  projection 


AMERICAN    HOUSE-CARPENTER. 


141 


in  comparison  with  its  height,  the  point,  e,  being  found  as  in  tho 
last  figure,  h  s  may  be  made  equal  to  5  e,  instead  of  o  s  as  in  toe 
last  figure.  Fig.  181  is  appropriate  for  a  crown  moulding  of  a 
cornice.  In  this  figure  the  height  and  projection  are  given ;  the 
direction  of  the  diameter,  a  b,  drawn  through  the  middle  cf 
the  diagonal,  e  /,  is  taken  at  pleasure ;  and  d  c  is  parallel  to  a 
e.  To  find  the  length  of  d  c,  draw  b  A,  at  right  angles  to  a  6  y 
upon  o,  with  o  f  for  radius,  describe  the  arc,/  A,  cutting  6  h  in 
h  ;  then  make  o  c  and  o  d,  each  equal  to  b  h.*  To  draw  the  curve, 
see  note  to  Art.  118.  Fig.  182  to  186  are  peculiarly  distinct  from 
ancient  mouldings,  being  composed  principally  of  straight  lines  ; 
the  few  curves  they  possess  are  quite  short  and  quick 


H.  P. 


H.P. 


Fig.  1ST. 


Fig.  1S8. 


250. — Fig.  187  and  188  are  designs  for   antae  caps.      The 

*  The  manner  of  ascertaining  the  length  of  the  conjugate  diameter,  dc,'m  this  figure, 
and  also  in  Fig.  143, 193  and  199  is  new,  and  is  important  in  this  application.  It  ia 
founded  upon  well-known  mathematical  principles,  viz :  All  the  parallelograms  that  may 
be  circumscribed  about  an  ellipsis  are  equal  to  one  another,  and  consequently  any  one 
is  equal  to  the  rectangle  of  the  two  axes.  And  again  :  the  sum  of  the  squares  of  cver>' 
pair  of  conjugate  diameters  is  eqxial  to  the  sum  of  the  squares  of  the  two  axes. 


142 


AMERICAN   HOUSE-CARPENTER. 


diameter  of  the  antae  is  divided  into  20  equal  parts,  and  the 
height  and  projection  of  the  members,  are  regulated  in  accord- 
ance with  those  parts,  as  denoted  under  If  and  P,  height  and 
projection.  The  projection  is  measured  from  the  middle  of 
the  antes.  These  will  be  found  appropriate  for  porticos,  door- 
ways, mantel-pieces,  door  and  window  trimmings,  <fec.  The 
height  of  the  antse  for-  mantel-pieces,  should  be  from  5  to  6 
diameters,  having  an  entablature  of  from  2  to  2£  diameters. 
This  is  a  good  proportion,  it  being  similar  to  the  Doric  order. 
But  for  a  portico  these  proportions  are  much  too  heavy ;  an 
antre,  15  diameters  high,  and  an  entablature  of  3  diameters, 
will  have  a  better  appearance. 


CORNICES. 


251. — Fig.  189  to  197  are  designs  for  eave  cornices,  and 
Fig.  198  and  199  are  for  stucco  cornices  for  the  inside  finish 
of  rooms.  In  some  of  these  the  projection  of  the  uppermost 
member  from  the  facia,  is  divided  into  twenty  equal  parts, 


MOULDINGS.    CORNICES.    &C. 


113 


and  the  various  members  are  proportioned  according  to  those 
parts,  as  figured  under  H  and  P. 


J 


uyi/^v ..;• .;  A- 1 


JJJJJLUUJDJUL1LIJUULIJ 


Fig.  191. 


144 


AMERICAN   HOUSE-CARPENTER. 


Fir  193. 


Fig.  183. 


MOULDINGS,    CORNICES,    &C. 


145 


Fig.  194. 


H.  P. 


Fig.  195. 

19 


II.  p. 


AMERICAN    HOUSE-CARPENTER. 


Fig.  196. 


Fig.  197. 


MOULDINGS,    COPvNICES,    &C. 


147 


Fig.  198. 


Fig.  199. 


148 


AMERICAN    HOUSE-CARPENTER. 

d 


b        123 

Fig.  200. 


252. — To  proportion  an  eave  cornice  in  accordance  with  the 
height  of  the  building.  Draw  the  line,  a  c,  (Fig.  200,)  and 
make  b  c  and  b  a,  each  equal  to  36  inches ;  from  6,  draw  b  d,  at 
right  angles  to  a  c,  and  equal  in  length  to  ^  of  a  c  ;  bisect  6  d  in 
e,  and  from  a,  through  e,  draw  a  f;  upon  a,  with  a  c  for  radius, 
describe  the  arc.  c/,  and  upon  e,  with  e/for  radius,  describe  the 
arc,/d;  divide  the  curve,  df  c,  into  7  equal  parts,  as  at  10,  20, 
30,  &c.,  and  from  these  points  of  division,  draw  lines  to  b  c,  pa- 
rallel to  d  b  ;  then  the  distance,  b  1,  is  the  projection  of  a  cornice 
for  a  building  10  feet  high  ;  b  2,  the  projection  at  20  feet  high  ; 
b  3,  the  projection  at  30  feet,  &c.  If  the  projection  of  a  cornice  for 
a  building  34  feet  high,  is  required,  divide  the  arc  between  30  and 
40  into  10  equal  parts,  and  from  the  fourth  point  from  30,  draw  a 
line  to  the  base,  b  c,  parallel  with  b  d  ;  then  the  distance  of  the 
point,  at  which  that  line  cuts  the  base,  from  6,  will  be  the  projec- 
tion required.  So  proceed  for  a  cornice  of  any  height  within  70 
feet.  The  above  is  based  on  the  supposition  that  36  inches  is  the 
proper  projection  for  a  cornice  70  feet  high.  This,  for  general 
purposes,  will  be  found  correct ;  still,  the  length  of  the  line,  b  c, 
may  be  varied  to  suit  the  judgment  of  those  who  think  differ- 
ently. 

Having  obtained  the  projection  of  a  cornice,  divide  it  into  20 
equal  parts,  and  apportion  the  several  members  according  to  its 
destination— as  is  shown  at  Fig.  195, 196,  and  197. 


MOULDINGS,    CORNICES,    &C. 


J 


Fig.  20L 

253. —  To  proportion  a  cornice  according  to  a  smaller  given 
one.  Let  the  cornice  at  Fig.  201  be  the  given  one.  Upon  any 
point  in  the  lowest  line  of  the  lowest  member,  as  at  a,  with  the 
height  of  the  required  cornice  for  radius,  describe  an  intersecting 
arc  across  the  uppermost  line,  as  at  b  /join  a  and  b  :  then  b  1  will 
oe  the  perpendicular  height  of  the  upper  fillet  for  the  proposed  cor- 
nice, 1  2  the  height  of  the  crown  moulding — and  so  of  all  the 
members  requiring  to  be  enlarged  to  the  sizes  indicated  on  this 
line.  For  the  projection  of  the  proposed  cornice,  draw  a  d,  at  right 
angles  to  a  6,  and  c  d,  at  right  angles  to  be;  parallel  with  c  d, 
draw  lines  from  each  projection  of  the  given  cornice  to  the  line, 
ad;  then  e  d  will  be  the  required  projection  for  the  proposed 
cornice,  and  the  perpendicular  lines  falling  upon  e  d  will  indicate 
the  proper  projection  for  the  members. 

254. —  To  proportion  a  cornice  according  to  a  larger  given 
one.  Let  A,  (Fig.  202,)  be  the  given  cornice.  Extend  a  o  to  b, 
and  draw  c  d,  at  right  angles  to  a  b  ;  extend  the  horizontal  lines 
of  the  cornice,  A,  until  they  touch  o  d ;  place  the  height  of  the 
proposed  cornice  from  o  to  e,  and  join  f  and  e  ;  upon  o,  with  the 
projection  of  the  given  cornice,  o  a,  for  radius,  describe  the  quad- 
rant, a  d  ;  from  d,  draw  d  b,  parallel  tofe;  upon  o,  with  o  b  for 
radius,  describe  the  quadrant,  be;  then  o  c  will  be  the  proper  pro- 
jection for  the  proposed  cornice.  Join  a  and  c  /  draw  lines  from  the 


150 


AMERICAN    HOUSE-CARPENTER. 


Fig.  202.         d 

projection  of  the  different  members  of  the  given  cornice  to  a  ot 
parallel  to  o  d  ;  from  these  divisions  on  the  line,  a  o,  draw  lines 
to  the  line,  o  c,  parallel  to  a  c  ;  from  the  divisions  on  the  line,  of, 
draw  lines  to  the  line,  o  e,  parallel  to  the  line,  /  e  ;  then  the  di- 
visions on  the  lines,  o  e  and  o  c,  will  indicate  the  proper  height  and 
projection  for  the  different  members  of  the  proposed  cornice.  In 
this  process,  we  have  assumed  the  height,  o  e,  of  the  proposed 
cornice  to  be  given ;  but  if  the  projection,  o  c,  alone  be  given,  we 
can  obtain  the  same  result  by  a  different  process.  Thus :  upon  o. 
with  o  c  for  radius,  describe  the  quadrant,  c  b  •  upon  o,  with  o  a 
for  radius,  describe  the  quadrant,  ad;  join  d  and  b  ;  from/,  draw 
f  e,  parallel  to  d  b  ;  then  o  e  will  be  the  proper  height  for  the  pro- 
posed cornice,  and  the  height  and  projection  of  the  different  mem- 
bers can  be  obtained  by  the  above  directions.  By  this  problem, 
a  cornice  can  be  proportioned  according  to  a  smaller  given  one 
as  well  as  to  a  larger  ;  but  the  method  described  in  the  previous 
article  is  much  more  simple  for  that  purpose. 

255. —  To  find  the  angle-bracket  for  a  cornice.  Let  A,  (Fig. 
203,)  be  the  wall  of  the  building,  and  B  the  given  bracket,  which, 
for  the  present  purpose,  is  turned  down  horizontally.  The  angle- 
bracket,  C,  is  obtained  thus :  through  the  extremity,  a,  and  paral- 


MOULDINGS,    CORNICES,    &C. 


151 


g      Fig.  20a 


Fig.  204. 


lei  with  the  wall,/rf,  draw  the  line,  a  b  ;  make  e  c  equal  a  /, 
and  through  c,  draw  c  6,  parallel  with  e  d  ;  join  d  and  6,  and  from 
the  several  angular  points  in  B,  draw  ordinates  to  cut  d  b  in  1,  2 
and  3 ;  at  those  points  erect  lines  perpendicular  to  d  b  ;  from  A, 
draw  h  g;  parallel  to/  a  ;  take  the  ordinates,  1  o,  2  o,  &c.,  at  B, 
and  transfer  them  to  C,  and  the  angle-bracket,  C,  will  be  defined. 
In  the  same  manner,  the  angle-bracket  for  an  internal  cornice,  or 
the  angle-rib  of  a  coved  ceiling,  or  of  groins,  as  at  Fig.  204,  can 
be  found. 

256. — A  level  crown  moulding  being  given,  to  find  the  raking 
moulding  and  a  level  return  at  the  top.  Let  A,  (Fig-  205,)  be 
the  given  moulding,  and  A  b  the  rake  of  the  roof.  Divide  the 
curve  of  the  given  moulding  into  any  number  of  parts,  equal  or 
unequal,  as  at  1,  2,  and  3  ;  from  these  points,  draw  horizontal 
•  lines  to  a  perpendicular  erected  from  c;  at  any  convenient  place 
on  the  rake,  as  at  B,  draw  a  c,  at  right  angles  to  A  b  ;  also,  from 
6,  draw  the  horizontal  line,  b  a  ;  place  the  thickness,  d  a,  of  the 
moulding  at  A,  from  b  to  a,  and  from  a,  draw  the  perpendicular 
line,  a  e  ;  from  the  points,  1,  2,  3,  at  A,  draw  lines  to  C,  parallel 
to  A  b  ;  make  a  1,  a  2  and  a  3,  at  B  and  at  C,  equal  to  a  1,  &c., 
at  A  ;  through  the  points,  1,  2  and  3,  at  B,  trace  the  curve — this 
will  be  the  proper  form  for  the  raking  moulding.  From  1,  2  and 


152 


AMERICAN    HOUSE-CARPENTER. 


Fig  205. 


3,  at  C,  drop  perpendiculars  to  the  corresponding  ordinates  from 
1,2  and  3,  at  ^/through  the  poir.ts  of  intersection,  trace  the 
curve — this  will  be  the  proper  form  lor  the  return  at  the  top. 


SECTION  I V.— FRAMING. 


257. — This  subject  is,  to  the  carpenter,  of  the  highest  impor- 
tance ;  and  deserves  more  attention  and  a  larger  place  in  a  volume 
ol  this  kind,  than  is  generally  allotted  to  it.  Something,  indeed, 
has  been  said  upon  the  geometrical  principles,  by  which  the  seve- 
ral lines  for  the  joints  and  the  lengths  of  timber,  may  be  ascer- 
tained ;  yet,  besides  this,  there  is  much  to  be  learned.  For  how- 
ever precise  or  workmanlike  the  joints  may  be  made,  what  will 
it  avail,  should  the  system  of  framing,  from  an  erroneous  position 
of  its  timbers,  &c.,  change  its  form,  or  become  incapable  of  sus- 
taining even  its  own  weight  ?  Hence  the  necessity  for  a  know- 
ledge of  the  laws  of  pressure  and  the  strength  of  timber.  These 
being  once  understood,  we  can  with  confidence  determine  the  best 
position  and  dimensions  for  the  several  timbers  which  compose  a 
floor  or  a  roof,  a  partition  or  a  bridge.  As  systems  of  framing 
are  more  or  less  exposed  to  heavy  weights  and  strains,  and,  in 
case  of  failure,  cause  not  only  a  loss  of  labour  and  material,  but 
frequently  that  of  life  itself,  it  is  very  important  that  the  materials 
employed  be  of  the  proper  quantity  and  quality  to  serve  their  des 
tination.  And,  on  the  other  hand,  any  superfluous  material  is  not 
only  useless,  but  a  positive  injury,  it  being  an  unnecessary  load 
upon  the  points  of  support.  It  is  necessary,  therefore,  to  know 

20 


154  AMERICAN    HOUSE-CARPENTER. 

the  least  quantity  of  timber  that  will  suffice  for  strength.  Tho 
greatest  fault  in  framing  is  that  of  using  an  excess  of  material. 
Economy,  at  least,  would  seem  to  require  that  this  evil  be  abated. 

Before  proceeding  to  considei  the  principles  upon  which  a  sys- 
tem of  framing  should  be  constructed,  let  us  attend  to  a  few  of 
the  elementary  laws  in  Mechanics,  which  will  be  found  to  be  of 
great  value  in  determining  those  principles. 

258. — LAWS  OF  PRESSURE.  (1.)  A  heavy  body  always 
exerts  a  pressure  equal  to  its  own  weight  in  a  vertical  direction. 
Example :  Suppose  an  iron  ball,  weighing  100  Ibs.,  be  supported 
upon  the  top  of  a  perpendicular  post,  (Fig.  220;)  then  the 
pressure  exerted  upon  that  post  will  be  equal  to  the  weight  of  the 
uall;  viz.,  100  Ibs.  (2.)  But  if  two  inclined  posts,  (Fig.  206,) 
be  substituted  for  the  perpendicular  support,  the  united  pressures 
upon  these  posts  will  be  more  than  equal  to  the  weight,  and  will 
be  in  proportion  to  their  position.  The  farther  apart  their  feet  are 
spread  the  greater  will  be  the  pressure,  anft  vice  versa.  Hence 
tremendous  strains  may  be  exerted  by  a  comparatively  small 
weight.  And  it  follows,  therefore,  that  a  piece  of  timber  intend- 
ed for  a  strut  or  post,  should  be  so  placed  that  its  axis  may  coin- 
cide, as  near  as  possible,  with  the  direction  of  the  pressure.  The 
direction  of  the  pressure  of  the  weight,  W,  (Fig.  206;)  is  in  the 
vertical  line,  b  d;  and  the  weight,  W,  would  fall  in  that  line,  if 
the  two  posts  were  removed,  hence  the  best  position  for  a  support 


Fig.  206. 


FRAMING.  1 55 

for  the  weight  would  be  in  that  line.  But,  as  it  rarely  occurs 
in  systems  of  framing  that  weights  can  be  supported  by  any 
single  resistance,  they  requiring  generally  two  or  more  sup- 
ports, (as  in  the  case  of  a  roof  supported  by  its  rafters,)  it  be 
comes  important,  therefore,  to  know  the  exact  amount  of  pres- 
sure any  certain  weight  is  capable  of  exerting  upon  oblique 
supports.  Now  it  has  been  ascertained  that  the  three  lines  of 
a  triangle,  drawn  parallel  with  the  direction  of  three  concur- 
ring forces  in  equilibrium,  are  in  proportion  respectively  to 
these  forces.  For  example,  in  Fig.  206,  we  have  a  represen- 
tation of  three  forces  concurring  in  a  point,  which  forces  are 
in  equilibrium  and  at  rest ;  thus,  the  weight,  TF",  is  one  force, 
and  the  resistance  exerted  by  the  two  pieces  of  timber  are  the 
other  two  forces.  The  direction  in  which  the  first  force  acts  is 
vertical — downwards  ;  the  direction  of  the  two  other  forces  is 
in  the  axis  of  each  piece  of  timber  respectively.  These  three 
forces  all  tend  towards  the  point,  b. 

Draw  the  axes,  a  b  and  b  c,  of  the  two  supports ;  make  I  d 
vertical,  and  from  d  draw  d  e  and  d  f  parallel  with  the  axes, 
b  c  and  b  a,  respectively.  Then  the  triangle,  b  d  e,  has  its 
lines  parallel  respectively  with  the  direction  of  the  three 
forces ;  thus,  b  d  is  in  the  direction  of  the  weight,  TT,  d  e 
parallel  with  the  axis  of  the  timber  b  <?,  and  e  b  is  in  the 
direction  of  the  timber  a  b.  In  accordance  with  the  principle 
above  stated,  the  lengths  of  the  sides  of  the  triangle,  b  d  e,  are 
in  proportion  respectively  to  the  three  forces  aforesaid  ;  thus — 

As  the  length  of  the  line,  I  d, 

Is  to  the  number  of  pounds  in  the  weight,  TF, 

So  is  the  length  of  the  line,  b  e, 

To  the  number  of  pounds'  pressure  resisted  by  the  timber, 

a  I. 
Again — 

As  the  length  of  the  line,  b  d, 

Is  to  the  number  of  pounds  in  the  weight,  Wt 


156  AMERICAN   HOUSE-CAKPENTEK. 

So  is  the  length  of  the  line,  d  e, 

To  the  number  of  pounds'  pressure  resisted  by  the  timber, 

be. 
And  again — 

As  the  length  of  the  line,  b  <?, 
Is  to  the  pounds'  pressure  resisted  by  a  b, 
So  is  the  length  of  the  line,  d  e, 
To  the  pounds'  pressure  resisted  by  b  c. 
These  proportions  are  more  briefly  stated  thus — 

1st.  bd  :  W::  be:  P, 

P  being  used  as  a  symbol  to  represent  the  number  of  founds' 
pressure  resisted  by  the  timber,  a  b. 

2nd.  bd  :  W ::  d  e  :  Q, 

Q  representing  the  number  of  pounds'  pressure  resisted  by  the 
timber,  b  c. 

3d.  be:P  ::  de:  Q. 

259. — This  relation  between  lines  and  pressures  is  important, 
and  is  of  extensive  application  in  ascertaining  the  pressures 
induced  by  known  weights  throughout  any  system"  of  framing. 
The  parallelogram,  b  e  d  f,  is  called  the  Parallelogram  of 
Forces  ;  the  two  lines,  b  e  and  b  /,  being  called  the  compo- 
nents, and  the  line  b  d  the  resultant.  Where  it  is  required  to 
find  the  components  from  a  given  resultant,  (Fig.  206,)  it  is 
not  needed  to  draw  the  fourth  line,  df,  for  the  triangle,  b  d  e, 
gives  the  desired  result.  But  when  the  resultant  is  to  be 
ascertained  from  given  components,  (Fig.  212,)  it  is  more  con- 
venient to  draw  the  fourth  line. 

260. — The  Resolution  of  Forces  is  the  finding  of  two  or 
more  forces,  which,  acting  in  different  directions,  shall  exactly 
balance  the  pressure  of  any  given  single  force.  To  make  a 
practical  application  of  this,  let  it  be  required  to  ascertain 
the  oblique  pressure  in  Fig.  206.  In  this  Fig.  the  line  b  d 
measures  half  an  inch,  (0*5  inch,)  and  the  line  b  e  three- 
tenths  of  an  inch,  (0*3  inch.)  Now  if  the  weight,  TF,  be  sup- 


FRAMING.  157 

posed  to  be  1200  pounds,  then  the  first  stated  proportion 
above, 

Id  :  W::be:P, 
becomes    • 

0-5  :  1200  ::  0-3  :  P. 

And  since  the  product  of  the  means  divided  by  one  of  the 
extremes  gives  the  other  extreme,  this  proportion  may  be  put 
in  the  form  of  an  equation,  thus  — 


0-5 


Performing  the  arithmetical  operation  here  indicated,  that  is, 
multiplying  together  the  two  quantities  above  the  line,  and 
dividing  the  product  by  the  quantity  under  the  line,  the  quo- 
tient will  be  equal  to  the  quantity  represented  by  P,  viz.,  the 
pressure  resisted  by  the  timber,  a  5.  Thus  — 

1200 
0-3 

0-5)360-0 

720  =  P. 

The  strain  upon  the  timber,  a  5,  is,  therefore,  equal  to  720 
pounds  ;  and  the  strain  upon  the  other  timber,  5  c  ,  is  also  720 
pounds;  for  in  this  case,  the  two  timbers  being  inclined 
equally  from  the  vertical,  the  line  e  d  is  therefore  equal  to  the 
line  &  *. 


Fig.  207. 


158  AMERICAN   HOUSE-CARPENTER. 

261.  —  In  Fig.  207,  the  two  supports  are  inclined  at  different 
angles,  and  the  pressures  are  proportionately  unequal.  The 
supports  are  also  unequal  in  length.  The  length  of  the  sup- 
ports does  not  alter  the  amount  of  pressure  from  the  concen- 
trated load  supported  ;  but  generally  long  timbers  are  not  so 
capable  of  resistance  as  shorter  ones.  They  yield  more  readily 
laterally,  as  they  are  not  so  stiff,  and  shorten  more,  as  the  com- 
pression is  in  proportion  to  the  length.  To  ascertain  the  pres- 
sures in  Fig.  207,  let  the  weight  suspended  from  b  d  be  equal 
to  two  and  three-quarter  tons,  (2*75  tons.)  The  line  5  d  mea- 
sures five  and  a  half  tenths  of  an  inch,  (0-55  inch,)  and  the  line 
I  e  half  an  inch,  (0*5  inch.)  Therefore,  the  proportion 
I  d  :  W  ::  b  e  :  P,  becomes  0-55  :  2-75  ::  0-5  :  P, 


j 

0'55 

2-75 
0-5 

0-55)1-375(2-5  =  P. 
110 

275 
275 

The  strain  upon  the  timber,  5  e,  is,  therefore,  equal  to  two 
and  a  half  tons. 

Again,  the  line  e  d  measures  four-tenths  of  an  inch,  (0*4 
inch  ;)  therefore,  the  proportion 

I  d  :  W  ::  e  d  :  Q,  becomes  0-55  :  2-75  ::  0-4  :  Q, 
,  2-75  x  0-4      ~ 


2-75 
0-4 

0-55)1-100(2  =  Q. 
110 


FKAMESG.  16S 

The  strain  upon  the  timber,  bf,  is,  therefore,  equal  to  two 
tons. 

262. — Thus  it  is  seen  that  the  united  pressures  exerted  by  a 
weight  upon  two  inclined  supports  always  exceed  the  weight. 
In  the  last  case  21  tons  exerts  a  pressure  of  2i  and  two  tons, 
equal  together  to  4£  tons;  "and  in  the  former  case,  1200 
pounds  exerts  a  pressure  of  twice  720  pounds,  equal  to  1440 
pounds.  The  smaller  the  angle  of  inclination  to  the  horizon- 
tal, the  greater  will  be  the  pressure  upon  the  supports.  So,  in 
the  frame  of  a  roof,  the  strain  upon  the  ratters  decreases  gra- 
dually with  the  increase  of  the  angle  of  inclination  to  the 
horizon,  the  length  of  the  rafter  remaining  the  same. 

263. — This  is  true  in  comparing  systems  of  framing  with 
each  other ;  but  in  a  system  where  the  concentrated  weight 
to  be  supported  is  not  in  the  middle,  (see  Fig.  207,)  and,  in 
consequence,  the  supports  are  not  inclined  equally,  the  strain 
will  be  greatest  upon  the  support  that  has  the  greatest  inclina- 
i»oT>  to  the  horizon. 

<I64. — In  ordinary  cases,  in  roofs  for  example,  the  load  is 
not  concentrated  but  is  that  of  the  framing  itself.  Here  the 
amount  of  the  load  will  be  in  proportion  to  the  length  of  the 
rafter,  and  the  rafter  increases  in  length  with  the  increase  of 
the  angle  of  inclination,  the  span  remaining  the  same.  So  it 
is  seen  that  in  enlarging  the  angle  of  inclination  to  the  horizon 
in  order  to  lessen  the  oblique  thrust,  the  load  is  increased  in 
consequence  of  the  elongation  of  the  rafter,  thus  increasing  the 
oblique  thrust.  Hence  there  is  a  limit  to  the  angle  of  inclina- 
tion. A  rafter  will  have  the  least  oblique  thrust  when  its 
angle  of  inclination  to  the  horizon  is  35°  16'  nearly.  This 
angle  is  attained  very  nearly  when  the  rafter  rises  8£  inches 
per  foot ;  or,  when  the  height,  B  C,  (Fig.  216,)  is  to  the  base, 
A  (7,  as  8J  is  to  12,  or  as  0-7071  is  to  1-0. 

265. — Correct  ideas  of  the  comparative  pressures  exerted 
mp'on  timbers,  according  to  their  position,  will  be  readily 


160  AMERICAN    HOUSE-CARPENTER. 

formed  by  drawing  various  designs  of  framing,  and  estimating 
the  several  strains  in  accordance  with  the  parallelogram  of 
forces,  always  drawing  the  triangle,  5  d  e,  so  that  the  three 
lines  shall  he  parallel  with  the  three  forces,  or  pressures,  re- 
spectively. The  length  of  the  lines  forming  this  triangle  is 
unimportant,  but  it  will  be  found  more  convenient  if  the  line 
drawn  parallel  with  the  known  force  is  made  to  contain  as 
many  inches  as  the  known  force  contains  pounds,  or  as  many 
tenths  of  an  inch  as  pounds,  or  as  many  inches  as  tons,  or 
tenths  of  an  inch  as  tons  :  or,  in  general,  as  many  divisions  of 
any  convenient  scale  as  there  are  units  of  weight  or  pressure 
in  the  known  force.  If  drawn  in  this  manner,  then  the  num- 
ber of  divisions  of  the  same  scale  found  in  the  other  two  lines 
of  the  triangle  will  equal  the  units  of  pressure  or  weight  of  the 
other  two  forces  respectively,  and  the  pressures  sought  will  be 
ascertained  simply  by  applying  the  scale  to  the  lines  of  the 
triangle. 

For  example,  in  Fig.  207,  the  vertical  line,  5  <?,  of  the  tri- 
angle, measures  fifty-five  hundredths  of  an  inch,  (0'55  inch  ;) 
the  line,  5  «,  fifty-hundred ths,  (0'50  inch ;)  and  the  line,  e  dt 
forty,  (0-40  inch.)    Now,  if  it  be  supposed  that  the  vertical  pres- 
sure, or  the  weight  suspended  below  T)  d,  is  equal  to  55  pounds, 
then  the  pressure  on  J  e  will  equal  50  pounds,  and  that  on  e  d 
will  equal  40  pounds  ;  for,  by  the  proportion  above  stated, 
b  d  :  W  ::  ~b  e  :  P, 
55  :  55  ::  50  :  50 ; 
and  so  of  the  other  pressure. 

266. — If  a  scale  cannot  be  had  of  equal  proportions  with  the 
forces,  the  arithmetical  process  will  be  shortened  somewhat  by 
making  the  line  of  the  triangle  that  represents  the  known 
weight  equal  to  unity  of  a  decimally  divided  scale,  then  the 
other  lines  will  be  measured  in  tenths  or  hundredths ;  and  in 
the  numerical  statement  of  the  proportions  between  the  lines 
and  forces,  the  first  term  being  unity,  the  fourth  term  will  be 


FRAMING. 


161 


ascertained  simply  by  multiplying  the  second  and  third  terms 
together. 

For  example,  if  the  three  lines  are  1,  O'Y  and  1*3,  and  the 
known  weight  is  6  tons,  then 

Id  :  W::l>e:P,  becomes 
1 :  6  ::  0-T  :  P  -  4-2, 
equals  four  and  two-tenths  tons.     Again — 

I  d  :  W ::  e  d  :  Q,  becomes 
1  :  6  ::  1-3  :  Q  =  7-8, 
equals  seven  and  eight-tenths  tons. 


Fig.  203. 


267.— In  Fig.  208  the  weight,  TF,  exerts  a  pressure  on  the 
struts  in  the  direction  of  their  length  ;  their  feet,  n  n,  have, 
therefore,  a  tendency  to  move  in  the  direction  n  0,  and  would 
so  move,  were  they  not  opposed  by  a  sufficient  resistance  from 
the  blocks,  A  and  A.  If  a  piece  of  each  block  be  cut  off  at 
the  horizontal  line,  a  n,  the  feet  of  the  struts  would  slide  away 
from  each  other  along  that  line,  in  the  direction,  n  a ;  but  if, 
instead  of  these,  two_  pieces  were  cut  off  at  the  vertical  line, 
n  5,  then  the  struts  would  descend  vertically.  To  estimate  the 
horizontal  and  the  vertical  pressures  exerted  by  the  struts,  let 
n  o  be  made  equal  (upon  any  scale  of  equal  parts)  to  the  num- 
21 


162 


AMERICAN   HOUSE-CARPENTER. 


bcr  of  tons  with  which  the  strut  is  pressed  ;  construct  the 
parallelogram  of  forces  by  drawing  o  e  parallel  to  a  n,  and  of 
parallel  tobn;  then  nf,  (by  the  same  scale,)  shows  the  num- 
ber of  tons  pressure  that  is  exerted  by  the  strut  in  the  direc- 
tion n  a,  and  n  e  shows  the  amount  exerted  in  the  direction 
n  1).  By  constructing  designs  similar  to  this,  giving  various 
and  dissimilar  positions  to  the  struts,  and  then  estimating  the 
pressures,  it  will  be  found  in  every  case  that  the  horizontal 
pressure  of  one  strut  is  exactly  equal  to  that  of  the  other,  how- 
ever much  one  strut  may  be  inclined  more  than  the  other ; 
and  also,  that  the  united  vertical  pressure  of  the  two  struts  is 
exactly  equal  to  the  weight,  W.  (In  this  calculation  the 
weight  of  the  timbers  has  not  been  taken  into  consideration, 
simply  to  avoid  complication  to  the  learner.  In  practice  it  is 
requisite  to  include  the  weight  of  the  framing  with  the  load 
upon  the  framing.) 


Fig.  209. 


268.— Suppose  that  the  two  struts,  B  and  JB,  (Fig.  208.) 
were  rafters  of  a  roof,  and  that  instead  of  the  blocks,  A  and  A- 
the  walls  of  a  building  were  the  supports :  then,  to  prevent 
the  walls  from  being  thrown  over  by  the  thrust  of  B  and  B, 
it  would  be  desirable  to  remove  the  horizontal  pressure.  Thia 


163 

may  be  done  by  uniting  the  feet  of  the  rafters  with  a  rope, 
iron  rod,  or  piece  of  timber,  as  In  Fig.  209.  This  figure  is 
similar  to  the  truss  of  a  roof.  The  horizontal  strains  on  the 
tie-beam,  tending  to  pull  it  asunder  in  the  direction  of  ltd 
length,  may  be  measured  at  the  foot  of  the  rafter,  as  was 
shown  at  Fig.  208 ;  but  it  can  be  more  readily  and  as  accu- 
rately measured,  by  drawing  from  f  and  e  horizontal  lines  to 
the  vertical  line,  I  d,  meeting  it  in  o  and  o  ;  then  f  o  will  be 
the  horizontal  thrust  at  J3,  and  e  o  at  A  /  these  will  be  found 
to  equal  one  another.  When  the  rafters  of  a  roof  are  thug 
connected,  all  tendency  to  thrust  the  walls  horizontally  is 
removed,  the  only  pressure  on  them  is  in  a  vertical  direction, 
being  equal  to  the  weight  of  the  roof  and  whatever  it  has  to 
support.  This  pressure  is  beneficial  rather  than  otherwise,  as 
a  roof  having  trusses  thus  formed,  and  the  trusses  well  braced 
to  each  other,  tends  to  steady  the  walls. 


Fig.  211. 


269.— J%.  210  and  211  exhibit  methods  of  framing  for  sup- 
porting the  equal  weights,  W  and  TF.     Suppose  it  be  required 


164  AMERICAN   HOUSE-CARPENTER. 

to  measure  and  compare  the  strains  produced  on  the  pieces, 
A  B  and  A  C.  Construct  the  parallelogram  of  forces,  e  bf  d, 
according  to  Art.  258.  Then  l)f  will  show  the  strain  on  A  B, 
and  b  e  the  strain  on  A  C.  By  comparing  the  figures,  b  d  be- 
ing equal  in  each,  it  will  be  seen  that  the  strains  in  Fig.  210 
are  about  three  times  as  great  as  those  in  Fig.  211 :  the  posi- 
tion of  the  pieces,  A  B  and  A  (7,  in  Fig.  211,  is  therefore  far 
preferable. 


C        Fig.  212. 

270. — The  Composition  of  Forces  consists  in  ascertaining  the 
direction  and  amount  of  one  force,  which  shall  be  just  capable 
of  balancing  two  or  more  given  forces,  acting  in  different 
directions.  This  is  only  the  reverse  of  the  resolution  of  forces, 
and  the  two  are  founded  on  one  and  the  same  principle,  and 
may  be  solved  m  the  same  manner.  For  example,  let  A  and 
B,  (Fig.  212,)  be  two  pieces  of  timber,  pressed  in  the  direction 
of  their  length  towards  b — A  by  a  force  equal  to  6  tons  weight, 
and  B  equal  to  9.  To  find  the  direction  and  amount  of  pres- 
sure they  would  unitedly  exert,  draw  the  lines,  b  e  and  bf,  in 
a  line  with  the  axes  of  the  timbers,  and  make  I  e  equal  to  the 
pressure  exerted  by  B,  viz.,  9 ;  also  make  b  f  equal  to  the 
pressure  on  A,  viz.,  6,  and  complete  the  parallelogram  of 
forces,  e  b  f  d  ;  then  b  d,  the  diagonal  of  the  parallelogram, 
will  be  the  direction,  and  its  length,  9'25,  will  be  the  amount^ 


FRAMING.  165 

of  the  united  pressures  of  A  and  of  B.  The  line,  I  d,  is 
termed  the  resultant  of  the  two  forces,  If  and  be.  If  A  and 
B  are  to  be  supported  by  one  post,  (7,  the  best  position  for 
that  post  will  be  in  the  direction  of  the  diagonal,  Id;  and  it 
will  require  to  be  sufficiently  strong  to  support  the  united 
pressures  of  A  and  pf  J?,  which  are  equal  to  9'25  or  9i  tons. 


Fig.  213. 


271. — Another  example  :  let  Fig.  213  represent  a  piece  of 
framing  commonly  called  a  crane,  which  is  used  for  hoisting 
heavy  weights  by  means  of  the  rope.  B  bf,  which  passes  over 
a  pulley  at  &.  This  is  similar  to  Fig.  210  and  211,  yet  it  is 
materially  different.  In  those  figures,  the  strain  is  in  one 
direction  only,  viz.,  from  5  to  d ;  but  in  this  there  are  two 
strains,  from  A  to  B  and  from  A  to  W.  The  strain  in  the 
direction  A  B  is  evidently  equal  to  that  in  the  direction  A  W. 
To  ascertain  the  best  position  for  the  strut,  A  C,  make  b  e 
equal  to  &/",  and  complete  the  parallelogram  of  forces,  e  bfd; 
then  draw  the  diagonal,  b  <7,  and  it  will  be  the  position  re- 
quired. Should  the  foot,  C",  of  the  strut  be  placed  either 
higher  or  lower,  the  strain  on  A  G  would  be  increased.  In 
constructing  cranes,  it  is  advisable,  in  order  that  the  piece, 
B  A,  may  be  under  a  gentle  pressure,  to  place  the  foot  of  the 


1C6 


AMERICAN   HOUSE-CARPENTER. 


strut  a  trifle  lower  than  where  the  diagonal,  I  d,  would  indi 
cate,  but  never  higher. 


272. — Ties  and  Struts.  Timbeiu  in  a  state  of  tension  are 
called  ties,  while  such  as  are  in  a  state  of  compression  are 
termed  struts.  This  subject  can  be  illustrated  in  the  following 
manner : 

Let  A  and  B,  (Fig.  214,)  represent  beams  of  timber  support- 
ing the  weights,  TF,  W  and  TFy  A  having  but  one  support, 
which  is  in  the  middle  of  its  length,  and  B  two,  one  at  each 
end.  To  show  the  nature  of  the  strains,  let  each  beam  be 
sawed  in  the  middle  from  a  to  J.  The  effects  are  obvious : 
the  cut  in  the  beam,  A,  will  open,  whereas  that  in  JS  will 
close.  If  the  weights  are  heavy  enough,  the  beam,  A,  will 
break  at  5;  while  the  cut  in  B  will  be  closed  perfectly  tight 
at  a,  and  the  beam  be  very  little  injured  by  it.  But  if,  on  the 
other  hand,  the  cuts  be  made  in  the  bottom  edge  of  the  tim- 
bers, from  o  to  5,  B  will  be  seriously  injured,  while  A  will 
scarcely  be  affected.  By  this  it  appears  evident  that,  in  a 
piece  of  timber  subject  to  a  pressure  across  the  direction  of  its 
length,  the  fibres  are  exposed  to  contrary  strains.  If  the  tim- 
ber is  supported  at  both  ends,  as  at  JS,  those  from  the  top  edge 
down  to  the  middle  are  compressed  in  the  direction  of  their 
length,  while  those  from  the  middle  to  the  bottom  edge  are  in 
a  state  of  tension ;  but  if  the  beam  is  supported  as  at  A,  the 
contrary  effect  is  produced  ;  while  the  fibres  at  the  middle  of 
either  beam  are  not  at  all  strained.  The  strains  in  a  framed 


FRAMING.  167 

truss  are  of  the  same  nature  as  those  in  a  single  beam.  The 
truss  for  a  roof,  being  supported  at  each  end,  has  its  tie-beam 
in  a  state  of  tension,  while  its  rafters  are  compressed  in  the 
direction  of  their  length.  By  this,  it  appears  highly  important 
that  pieces  in  a  state  of  tension  should  be  distinguished  from 
such  as  are  compressed,  in  order  that  the  former  may  be  pre- 
served continuous.  A  strut  may  be  constructed  of  two  or 
more  pieces  ;  yet,  where  there  are  many  joints,  it  will  not 
resist  compression  so  well. 

273. — To  distinguish  ties  from  struts.  This  may  be  done 
by  the  following  rule.  In  Fig.  206,  the  timbers,  a  I  and  5  <?, 
are  the  sustaining  forces,  and  the  weight,  W,  is  the  straining 
force;  and,  if  the  support  be  removed,  the  straining  force 
would  move  from  the  point  of  support,  5,  towards  d.  Let  it  be 
required  to  ascertain  whether  the  sustaining  forces  are  stretched 
or  pressed  by  the  straining  force.  Rule :  upon  the  direction 
of  the  straining  force,  5  d,  as  a  diagonal,  construct  a  parallelo- 
gram, e  T)f  d,  whose  sides  shall  be  parallel  with  the  direction 
of  the  sustaining  forces,  a  Z>  and  c  d  ;  through  the  point,  5, 
draw  a  line,  parallel  to  the  diagonal,  ef;  this  may  then  bo 
called  the  dividing  line  between  ties  and  struts.  Because  all 
those  supports  which  are  on  that  side  of  the  dividing  line, 
which  the  straining  force  would  occupy  if  unresisted,  are  com- 
pressed, while  those  on  the  other  side  of  the  dividing  line  are 
stretched. 

In  Fig.  206,  the  supports  are  both  compressed,  being  on 
that  side  of  the  dividing  line  which  the  straining  force  would 
occupy  if  unresisted.  In  Fig.  210  and  211,  in  which  A  I>  and 
A  C  are  the  sustaining  forces,  A  G  is  compressed,  whereas 
A  B  is  in  a  state  of  tension ;  A  C  being  on  that  side  of  the 
line,  h  i,  which  the  straining  force  would  occupy  if  unresisted, 
and  A  E  on  the  opposite  side.  The  place  of  the  latter  might 
be  supplied  by  a  chain  or  rope.  In  Fig.  209,  the  foot  of  the 
rafter  at  A  is  sustained  by  two  forces,  the  wall  and  the  tie- 


168 


AMERICAN   HOUSE-CARPENTER. 


beam,  one  perpendicular  and  the  other  horizontal :  the  direc- 
tion of  the  straining  force  is  indicated  by  the  line,  5  a.  The 
dividing  line,  h  i,  ascertained  by  the  rule,  shows  that  the  wall 
is  pressed  and  the  tie-beam  stretched. 


Fig.  215. 

274.— Another  example  :  let  E A  B  F,  (Fig.  215,)  represent 
a  gate,  supported  by  hinges  at  A  and  E.  In  this  case,  the 
straining  force  is  the  weight  of  the  materials,  and  the  direction 
of  course  vertical.  Ascertain  the  dividing  line  at  the  several 
points,  G,  B,  f,  J,  H  and  F.  It  will  then  appear  that  the 
force  at  G  is  sustained  by  A  G  and  G  E,  and  the  dividing 
line  shows  that  the  former  is  stretched  and  the  latter  com- 
pressed. The  force  at  H'\%  supported  by  A  JJand  HE— the 
former  stretched  and  the  latter  compressed.  The  force  at  B 
is  opposed  by  H  £  and  A  B,  one  pressed,  the  other  stretched. 
The  force  at  F  is  sustained  by  G  F  and  F  E,  G  F  being 
stretched  and  F  E  pressed.  By  this  it  appears  that  A  B  is  in 
a  state  of  tension,  and  E  F,  of  compression  ;  also,  that  A  H 
and  G  F  are  stretched,  while  B  H  and  G  E  QXQ  compressed  : 
which  shows  the  necessity  of  having  A  II  and  G  F,  each  in 
one  whole  length,  while  B  H  and  G  E  may  be,  as  they  are 
shown,  each  in  two  pieces.  The  force  at  J  is  sustained  by 
G  J"and  J II,  the  former  stretched  arid  the  latter  compressed, 


FKAMING.  169 

The  piece,  C  D,  is  neither  stretched  nor  pressed,  and  conl  1  be 
dispensed  with  if  the  joinings  at  J  and  /  could  be  made  as 
effectually  without  it.  In  case  A  B  should  fail,  then  C  D 
would  be  in  a  state  of  tension. 

275.  —  The  centre  of  gravity.    The  centre  of  gravity  of  a 
uniform  prism  or  cylinder,  is  in  its  axis,  at  the  middle  of  its 
length  ;  that  of  a  triangle,  is  in  a  line  drawn  from  one  angle  to 
the  middle  of  the  opposite  side  and  at  one-third  of  the  length 
of  the  line  from  that  side  ;  that  of  a  right-angled  triangle,  at  a 
point  distant  from  the  perpendicular  equal  to  one-third  of  the 
base,  and  distant  from  the  base  equal  to  one-third  of  the  per- 
pendicular ;  that  of  a  pyramid  or  cone,  in  the  axis  and  at  one- 
quarter  of  the  height  from  the  base. 

276.  —  The  centre  of  gravity  of  a  trapezoid,  (a  four-sided 
figure  having  only  two  of  its  sides  parallel,)  is  in  a  line  joining 
the  centres  of  the,  two  parallel  sides,  and  at  a  distance  from 
the  longest  of  the  parallel  sides  equal  to  the  product  of  the 
length  into  the  sum  of  twice  the  shorter  added  to  the  longer 
of  the  parallel  sides,  divided  by  three  times  the  sum  of  the 
two  parallel  sides.     Algebraically  thus  — 

I  (2  a  +  V) 
=   3  (a  +  6) 

where  d  equals  the  distance  from  the  longest  of  the  parallel 
bides,  I  the  length  of  the  line  joining  the  two  parallel  sides, 
and  a  the  shorter  and  b  the  longer  of  the  parallel  sides. 

Example.—  A  rafter,  25  feet  long,  has  the  larger  end  14 
inches  wide,  and  the  smaller  end  10  inches  wide,  how  far  from 
the  larger  end  is  the  centre  of  gravity  located  ? 
Here,  I  -  25,  a  =  H,  and  5  =  if, 


25  X  34:      850 

Q      9^  =~72 

In  irregular  bodies  with  plain  sides,  the  centre  of  gravity 

22 


170 


AMERICAN   HOUSE-CARPENTER. 


may  be  found  by  balancing  them  upon  the  edge  of  a  prism— 
upon  the  edge  of  a  table — in  two  positions,  making  a  line  each 
time  upon  the  body  in  a  line  with  the  edge  of  the  prism,  and 
the  intersection  of  those  lines  will  indicate  the  point  required. 
Or  suspend  the  article  by  a  cord  or  thread  attached  to  one 
corner  or  edge ;  also,  from  the  same  point  of  suspension,  hang 
a  plumb-line,  and  mark  its  position  on  the  face  of  the  article ; 
again,  suspend  the  article  from  another  corner  or  side,  (nearly 
at  right  angles  to  its  former  position,)  and  mark  the  position 
of  the  plumb-line  upon  its  face ;  then  the  intersection  of  the 
two  lines  will  be  the  centre  of  gravity. 


U 


Fig.  216. 


277. — The  effect  of  the  weight  of  inclined  beams.  An  in- 
clined post  or  strut,  supporting  some  heavy  pressure  applied  at 
its  upper  end,  as  at  Fig.  209,  exerts  a  pressure  at  its  foot  in 
the  direction  of  its  length,  or  nearly  so.  But  when  such  a 
beam  is  loaded  uniformly  over  its  whole  length,  as  the  rafter 
of  a  roof,  the  pressure  at  its  foot  varies  considerably  from  the 
direction  of  its  length.  For  example,  let  A  B,  (Fig.  216,)  be 
a  beam  leaning  against  the  wall,  B  c,  and  supported  at  its 
foot  by  the  abutment,  A,  in  the  beam,  A  c,  and  let  o  be  the 
centre  of  gravity  of  the  beam.  Through  o,  draw  the  vertical 
line,  b  dy  and  from  B,  draw  the  horizontal  line,  B  &,  cutting 
b  d  in  b  •  join  b  and  J.,  and  b  A  will  be  the  direction  of  the 
thrust.  To  prevent  the  beam  from  loosing  its  footing,  the  joint 
at  A  should  be  made  at  right  angles  to  b  A.  The  amount  of 
pressure  will  be  found  thus :  let  b  d,  (by  any  scale  of  equal 


FRAMING.  171 

jaits,)  equal  the  number  of  tons  upon  the  beam,  A  B  j  draw 
d  e,  parallel  to  B  I ;  then  I  e,  (by  the  same  scale,)  equals  the 
pressure  in  the  direction,  5  A  ;  and  e  d,  the  pressure  against 
the  wall  at  B — and  also  the  horizontal  thrust  at  A,  as  these 
are  always  equal  in  a  construction  of  this  kind. 

278. — The  horizontal  thrust  of  an  inclined  beam,  (Fig.  216,) 
— the  effect  of  its  own  weight — may  be  calculated  thus  : 

Rule. — Multiply  the  weight  of  the  beam  in  pounds  by  its 
base,  A  C,  in  feet,  and  by  the  distance  in  feet  of  its  centre  of 
gravity,  o,  (see  Art.  275  and  276,)  from  the  lower  end,  at  A  ; 
and  divide  this  product  by  the  product  of  the  length,  A  B, 
into  the  height,  B  O,  and  the  quotient  will  be  the  horizontal 

thrust  in  pounds.     This  may  be  stated  thus  :  H  =  -  -     ,  where 

fi  I 

d  equals  the  distance  of  the  centre  of  gravity,  0,  from  the 
lower  end ;  5  equals  the  base,  A  0  j  w  equals  the  weight  of 
the  beam  ;  h  equals  the  height,  B  C j  I  equals  the  length  of 
the  beam ;  and  JT^quals  the  horizontal  thrust. 

Example. — A  beam,  20  feet  long,  weighs  300  pounds;  its 
centre  of  gravity  is  at  9  feet  from  its  lower  end ;  it  is  so 
inclined  that  its  base  is  16  feet  and  its  height  12  feet ;  what  is 
the  horizontal  thrust  ? 

a-       d  b  w  ,  9*x  16  x  300     9  x  4=  x  25 

Here  -j-j-  becomes  — = =  9x4x5 

hi  12  x  20  5 

=  180  =  H=  the  horizontal  thrust. 

This  rule  is  for  cases  where  the  centre  of  gravity  does  not 
occur  at  the  middle  of  the  length  of  the  beam,  although  it  is 
applicable  when  it  does  occur  at  the  middle ;  yet  a  shorter 
rule  will  suffice  in  this  case, — and  it  is  thus : — 

Rule. — Multiply  the  weight  of  the  rafter  in  pounds  by  the 
base,  A  C,  (Fig.  216,)  in  feet,  and  divide  the  product  by  twice 
the  height,  B  0,  in  feet ;  and  the  quotient  will  be  the  horizon 
tal  thrust,  when  the  cer  tre  of  gravity  occurs  at  the  middle  of 
the  beam. 


172 


AMERICAN   HOUSE-CAKPENTER. 


If  the  inclined  beam  is  loaded  with  an  equally  distributed 
load,  add  this  load  to  the  weight  of  the  beam,  and  use  this 
total  weight  in  the  rule  instead  of  the  weight  of  the  beam. 
And  generally,  if  the  centre  of  gravity  of  the  combined 
weights  of  the  beam  and  load  does  not  occur  at  the  centre  of 
the  length  of  the  beam  then  the  former  rule  is  to  be  used. 


Fig.  217. 


279. — In  Fig.  217,  two  equal  beams  are  supported  at  their 
feet  by  the  abutments  in  the  tie-beam.  This  case  is  similar  to 
the  last ;  for  it  is  obvious  that  each  beam  is  in  precisely  the 
position  of  the  beam  in  Fig.  216.  The  horizontal  pressures  at 
B,  being  equal  and  opposite,  balance  one  another ;  and  their 
horizontal  thrusts  at  the  tie-beam  are  also  equal.  (See  Art. 
2Q8—Fig.  209.)  When  the  height  of  a  roof,  (Fig.  217,)  is 
one-fourth  of  the  span,  or  of  a  shed,  (Fig.  216,)  is  one-half  the 
span,  the  horizontal  thrust  of  a  rafter,  whose  centre  of  gravity 
is  at  the  middle  of  its  length,  is  exactly  equal  to  the  weight 
distributed  uniformly  over  its  surface. 


fi«    218. 


FRAMING.  173 

280. — In  shed,  or  lean-to  roofs,  as  Fig.  216,  fie  horizontal 
pressure  will  be  entirely  removed,  if  the  bearings  of  the  raft- 
ers, as  A  B,  (Fig.  218,)  are  made  horizontal — provided,  how- 
ever, that  the  rafters  and  other  framing  do  not  bend  between 
the  points  of  support.  If  a  beam  or  rafter  have  a  natural 
curve,  the  convex  or  rounding  edge  should  be  laid  uppermost. 

281. — A  beam  laid  horizontally,  supported  at  each  end  and 
uniformly  loaded,  is  subject  to  the  greatest  strain  at  the  mid- 
dle of  its  length.  Hence  mortices,  large  knots  and  other  de- 
fects, should  be  kept  as  far  as  possible  from  that  point ;  and, 
in  resting  a  load  upon  a  beam,  as  a  partition  upon  a  floor 
beam,  the  weight  should  be  so  adjusted,  if  possible,  that  it  will 
bear  at  or  near  the  ends. 

Twice  the  weight  that  will  break  a  beam,  acting  at  the 
centre  of  its  length,  is  required  to  break  it  when  equally  dis- 
tributed over  its  length  ;  and  precisely  the  same  deflection  or 
sag  will  be  produced  on  a  beam  by  a  load  equally  distributed, 
that  five-eighths  of  the  load  will  produce  if  acting  at  the  centre 
of  its  length. 

282. — When  a  beam,  supported  at  each  end  on  horizontal 
bearings,  (the  beam  itself  being  either  horizontal  or  inclined,) 
has  its  load  equally  distributed,  the  amount  of  pressure  caused 
by  the  load  on  each  point  of  support  is  equal  to  one  half  the 
load  ;  and  this  is  also  the  case,  when  the  load  is  concentrated 
at  the  middle  of  the  beam,  or  has  its  centre  of  gravity  at  the 
middle  of  the  beam ;  but,  when  the  load  is  unequally  distri- 
buted or  concentrated,  so  that  its  centre  of  gravity  occurs  at 
some  other  point  than  the  middle  of  the  beam,  then  the  amount 
of  pressure  caused  by  the  load  on  one  -of  the  points  of  support 
is  unequal  to  that  on  the  other.  The  precise  amount  on  each 
may  be  ascertained  by  the  following  rule. 

Rule.— Multiply  the  weight  w,  (Fig.  219,)  by  its  distance,  C B, 
from  its  nearest  point  of  support,  B,  and  divide  the  product 
by  the  length,  A  I>,  of  the  beam,  and  the  quotient  wil  be  the 


174  AMERICAN   HOUSE-CARPENTER. 


Pig.  219. 

amount  of  pressure  on  the  remote  point  of  support,  A.  Again, 
deduct  this  amount  from  the  weight,  w,  and  the  remainder 
will  be  the  amount  of  pressure  on  the  near  point  of  support, 
B ;  or,  multiply  the  weight,  10,  by  its  distance,  A  C>  from  the 
remote  point  of  support,  A,  and  divide  the  product  by  the 
length,  A  B,  and  the  quotient  will  be  the  amount  of  pressure 
on  the  near  point  of  support,  B. 

When  I  equals  the  length,  A  B;  a  =  A  C;  I  =  0 B,  and 
w  =  the  load,  then 
«„  T. 

=  A  =  the  amount  of  pressure  at  A,  and 

w  ct/ 

—j—  =  B  =  the  amount  of  pressure  at  B. 

JExample. — A  beam.  20  feet  long  between  the  bearings,  has 
a  load  of  100  pounds  concentrated  at  3  feet  from  one  of  the 
bearings,  what  is  the  portion  of  this  weight  sustained  by  each 
bearing  ? 

Here  w  =  100  ;  a.  17 ;  5,  3 ;  and  Z,  20. 

w  I      100  x  3 
Hence  A  —— j—  =  — ™ —  =  15. 


w  a     100  x  17 

=-r=— 20— =85- 

Load  on  A  •=.  15  pounds. 
Load  on  B  =  85  pounds. 
Total  weight  =  If  0  pounds. 


1T5 


RESISTANCE    OF   MATEKIAL8. 

283. — Before  a  roof  truss,  or  other  piece  cf  framing,  can  be 
properly  designed,  two  things  are  required  to  be  known.  The 
one  is,  the  effect  of  gravity  acting  upon  the  various  parts  of 
the  intended  structure ;  the  other,  the  power  of  resistance 
possessed  by  the  materials  of  which  the  framing  is  to  be  con- 
structed. In  the  preceding  pages,  the  former  subject  having 
been  treated  of,  it  remains  now  to  call  attention  to  the  latter. 

284. — Materials  used  in  construction  are  constituted  in  their 
structure  either  of  fibres  (threads)  or  of  grains,  and  are  termed, 
the  former  fibrous,  the  latter  granular.  All  woods  and  wrought 
metals  are  fibrous,  while  cast  iron,  stone,  glass,  &c.,  are  gra- 
nular. The  strength  of  a  granular  material  lies  in  the  power 
of  attraction,  acting  among  the  grains  of  matter  of  which  the 
material  is  composed,  by  which  it  resists  any  attempt  to  sepa- 
rate its  grains  or  particles  of  matter.  A  fibre  of  wood  or  of 
wrought  metal  has  a  strength  by  which  it  resists  being  com- 
pressed or  shortened,  and  finally  crushed ;  also  a  strength  by 
which  it  resists  being  extended  or  made  longer,  and  finally 
sundered.  There  is  another  kind  of  strength  in  a  fibrous  mate- 
rial ;  it  is  the  adhesion  of  one  fibre  to  another  along  their  sides, 
or  the  lateral  adhesion  of  the  fibres. 

285. — In  the  strain  applied  to  a  piece  of  timber,  as  a  post 
supporting  a  weight  imposed  upon  it,  (Fig.  220,)  we  have  an 
instance  of  an  attempt  to  shorten  the  fibres  of  which  the  tim- 
ber is  composed.  The  strength  of  the  timber  in  this  case  is 
termed  the  resistance  to  compression.  In  the  strain  on  a  piece 
of  timber  like  a  king-post  or  suspending  piece,  ( J.,  Fig.  221,) 
we  have  an  instance  of  an  attempt  to  extend  or  lengthen  the 
fibres  of  the  material.  The  strength  here  exhibited  is  termed 
the  resistance  to  tension.  When  a  piece  of  timber  is  strained 
like  a  floor  beam,  or  any  horizontal  piece  carrying  a  load, 
(Fig.  222,)  we  have  an  instance,  in  which  the  two  strains  of 


176 


AMERICAN   HOUSE-CARPENTER. 


Fig.  222. 

compression  and  tension  are  brought  into  action ;  the  fibres  of 
the  upper  portion  of  the  beam  being  compressed,  and  those  of 
the  under  part  being  stretched.  This  kind  of  strength  of  tim 
ber  is  termed  resistance  to  cross  strains.  In  each  of  these  three 
kinds  of  strain  to  which  timber  is  subjected,  the  power  of 
resistance  is  in  a  measure  due  to  the  lateral  adhesion  of  the 
fibres,  not  so  much  perhaps  in  the  simple  tensile  strain,  yet  to 
a  considerable  degree  in  the  compressive  and  cross  strains. 
But  the  power  of  timber,  by  which  it  resists  a  pressure  acting 
compressively  in  the  direction  of  the  length  of  the  fibres,  tend- 
ing to  separate  the  timber  by  splitting  off  a  part,  as  in  the 
case  of  the  end  of  a  tie  beam,  against  which  the  foot  of  the 
rafter  presses — is  wholly  due  to  the  lateral  adhesion  of  the 
fibres. 

286. — The  strength  of  materials  is  that  power  by  which  they 
resist  fracture,  while  the  stiffness  of  materials  is  that  quality 
which  enables  them  to  resist  deflection  or  sagging.  A  know- 
ledge of  their  strength  is  useful,  in  order  to  determine  their 


FRAMING.  177 

Jimits  of  size  to  sustain  given  weights  safely ;  but  a  knowledge 
of  their  stiffness  is  more  important,  as  in  almost  all  construe 
tions  it  is  desirable  not  only  that  the  load  be  safely  sustained, 
but  that  no  appearance  of  weakness  be  manifested  by  any  sen- 
sible deflection  or  sagging. 

I. — RESISTANCE   TO   COMPRESSION. 

287. — The  resistance  of  materials  to  the  force  of  compression 
may  be  considered  in  four  several  ways,  viz. : 

1st.  "When  the  pressure  is  applied  to  the  fibres  longitudi 
nally,  and  on  short  pieces. 

2d.  When  the  pressure  is  applied  to  the  fibres  longitudi- 
nally, and  on  long  pieces. 

3d.  When  the  pressure  is  applied  to  the  fibres  longitudi- 
nally, and  so  as  to  split  off  the  part  pressed  against,  causing 
the  fibres  to  separate  by  sliding. 

4th.  When  the  pressure  is  applied  to  the  fibres  trans- 
versely. 

Posts  having  their  height  less  than  ten  times  their  least  side 
will  crush  before  bending ;  these  belong  to  the  first  case : 
while  posts,  whose  height  is  ten  times  their  least  side,  or  more 
than  ten  times,  will  bend  before  crushing ;  these  belong  to  the 
second  case. 

288. — In  the  above  first  and  fourth  cases  of  compression, 
experiment  has  shown  that  the  resistance  is  in  proportion  to 
the  number  of  fibres  pressed,  that  is,  in  proportion  to  the  area. 
For  example,  if  5,000  pounds  is  required  to  crush  a  prism  with 
a  base  1  inch  square,  it  will  require  20jOOO  pounds  to  crush  a 
prism  having  a  base  of  2  by  2  inches,  equal  to  4  inches  area : 
because  4  times  5,000  equals  20,000.  Experiment  has  also 
shown  that,  in  the  third  case,  the  resistance  is  in  proportion' to 
the  area  of  the  surface  separated  without  regard  to  the  form 
of  the  surface. 

289. — In  the  second  case  of  compression,  the  resistance  is  in 
23 


178  AMERICAN    HOUSE-CARPENTEE, 

proportion  to  the  area  of  the  cross  section  of  the  piece,  multi 
plied  by  the  square  of  its  thickness,  and  inversely  in  propor- 
tion to  the  square  of  the  length,  multiplied  by  the  weight. 
When  the  piece  is  square,  it  will  bend  and  break  in  the  direc- 
tion of  its  diagonal ;  here,  the  resistance  is  in  proportion  to  the 
square  of  the  diagonal  multiplied  by  the  square  of  the  dia- 
gonal, and  inversely  proportional  to  the  square  of  the  length 
multiplied  by  the  weight.  If  the  piece  is  round  or  cylindrical, 
its  resistance  will  be  in  accordance  with  the  square  of  the  dia- 
meter multiplied  by  the  square  of  the  diameter,  and  inversely 
proportional  to  the  square  of  the  length,  multiplied  by  the 
weight. 

290. — These  relations  between  the  dimensions  of  the  piece 
strained  and  its  resistance,  have  resulted  from  the  discussion 
of  the  subject  by  various  authors,  and  rules  based  upon  these 
relations  are  in  general  use,  yet  their  accuracy  is  not  fully 
established.  Some  experiments,  especially  those  by  Prof. 
Hoclgkinson,  have  shown  that  the  resistance  is  in  proportion  to 
a  less  power  of  the  diameter,  and  inversely  to  a  less  power  of 
the  height ;  yet  the  variance  is  not  great,  and  inasmuch  as  the 
material  is  restricted  in  the  rules  to  a  strain  decidedly  within 
its  limits  of  resistance,  no  serious  error  can  be  made  in  the 
use  of  rules  based  on  the  aforesaid  relations. 

291. — Experiments.  In  the  investigation  of  the  laws  appli- 
cable to  the  resistance  of  materials,  only  such  of  the  relations 
of  the  parts  have  been  considered  as  apply  alike  to  wood  and 
metal,  stone  and  glass,  or  other  material,  leaving  to  experi- 
ment the  task  of  ascertaining  the  compactness  and  cohesion  of 
particles,  and  the  tenacity  and  adhesion  of  fibres ;  those  quali- 
ties upon  which  depend  the  superiority  of  one  kind  of  material 
over  another,  and  which  is  represented  in  the  rules  by  a  constant 
number,  each  specific  kind  of  material  having  its  own  special 
constant^  obtained  by  experimenting  on  specimens  of  that 
peculiar  material. 


FRAMING. 


179 


292. — The  following  table  exhibits  the  results  of  experiments 
on  such  woods  as  are  in  most  common  use  in  this  country  for 
the  purpose  of  construction.  The  resistance  of  timber'  of  the 


TABLE   I. COiTPKESSIOST. 


Kind  of  Material. 

1 

1 

1 

1 

Pi 

of  //In  the 

J8. 

sh  Fibres  trans- 
clyja-lnch 

*if 

! 

£U 

1^ 

III 

PH 

|1 

111 
E-i 

|ll 

Pounds 

Pounds 

Pounds 

per  In. 

per  in. 

per  in. 

White  wood.      . 

•397 

•2432 

600 

600 

300 

Mahogany  (Bay  wood), 

•439 

3527 

880 

1300 

650 

Ash,  . 

•517 

4175 

1040 

2300 

1150 

Spruce, 

•369 

4199 

1050 

470 

160 

500 

250 

Chestnut,  . 

•491 

4791 

1200 

690 

230 

950 

475 

White  pino, 

•388 

4806 

1200 

490 

160 

600 

300 

Ohio  pine. 

•586 

4809 

1  200 

388 

130 

1250 

625 

Oak,  . 

•612 

5316 

1330 

780 

260 

1900 

950 

Hemlock,  . 

•423 

5400 

1350 

540 

180 

600 

300 

Black  walnut,- 

•421 

5594 

1400 

1600 

800 

Maple, 

•574 

6061 

1515 

2050 

1025 

Cherry,      . 

•494 

6477 

1620 

1900 

950 

White  oak, 

•774 

6660 

1665 

2000 

1000 

Georgia  pine, 

•613 

6767 

1700 

510 

170 

1700 

850 

Locust, 

•762 

7652 

1910 

1180 

400 

2100 

1050 

Live  oak,  . 

•916 

7936 

1980 

5100 

2550 

Mahogany  (Si  Domingo), 

•837 

8280 

2070 

4300 

2150 

Lignum  vitae, 

1'282 

8650 

2160 

5800 

2900 

Hickory,   . 

•877 

9817 

2450 

3100 

1550 

same  name  varies  much  ;  depending  as  it  obviously  must  on 
the  soil  in  which  it  grew,  on  its  age  before  and  after  cutting, 
on  the  time  of  year  when  cut,  and  on  the  manner  in  which  it 
has  been  kept  since  it  was  cut.  And  of  wood  from  the  same 
tree,  much  depends  upon  its  location,  whether  at  the  butt  or 
towards  the  limbs,  and  whether  at  the  heart  or  at  the  sap,  or 
at  a  point  midway  from  the  centre  to  the  circumference  of  the 
tree.  The  pieces  submitted  to  experiment  were  of  ordinary 
good  quality,  such  as  would  be  deemed  proper  to  be  used  in 
framing.  The  prisms  crushed  were  2  inches  long,  and  from  1 
inch  to  1 J  inches  square  ;  some  were  wider  one  way  than  the 


180  AMERICAN   HOUSE-CARPENTER. 

other,  Lnt  all  containing  in  area  of  cross  section  from  1  to  2 
inches.  There  were  generally  three  specimens  of  each  kind. 
The  weight  given  in  the  table  is  the  average  crushing  weight 
per  superficial  inch. 

In  the  preceding  table  the  first  column  contains  the  specific 
gravity  of  the  several  kinds  of  \vood,  showing  their  compara- 
tive density.  The  weight  in  pounds  of  a  cubic  foot  of  any 
kind  of  wood  or  other  material,  is  equal  to  its  specific  gravity 
multiplied  by  62'5  ;  this  number  being  the  weight  in  pounds 
of  a  cubic  foot  of  water.  The  second  column  contains  the 
weight  in  pounds  required  to  crush  a  prism  having  a  base  of 
one  inch  square  ;  the  pressure  applied  to  the  fibres  longitudi- 
nally. The  third  column  contains  the  value  of  (7  in  the  rules ; 
C  being  equal  to  one-fourth  of  the  crushing  weight  in  the 
preceding  column.  The  fourth  column  contains  the  weight 
in  pounds,  which,  applied  to  the  fibres  longitudinally,  is 
required  to  force  off  a  part  of  the  piece,  causing  the  fibres  to 
separate  by  sliding,  the  surface  separated  being  one  inch 
square.  The  fifth  column  contains  the  value  of  II  in  the 
rules,  II  being  equal  to  one  third  of  the  weight  in  the  preced- 
ing column.  The  sixth  column  contains  the  weight  in  pounds 
required  to  crush  the  piece  when  the  pressure  is  applied  to  the 
fibres  transversely,  the  piece  being  one  inch  thick,  and  the 
surface  crushed  being  one  inch  square,  and  depressed  one 
twentieth  of  an  inch  deep.  The  seventh  column  contains  the 
value  of  P  in  the  rules ;  P  being  the  weight  in  pounds  applied 
to  the  fibres  transversely,  which. is  required  to  make  a  sensible 
impression  one  inch  square  on  the  side  of  the  piece,  this  being 
the  greatest  weight  that  would  be  proper  for  a  post  to  be 
loaded  with  per  inch  surface  of  bearing,  resting  on  the  side  of 
the  kind  of  wood  set  opposite  in  the  table.  A  greater  weight 
would,  in  proportion  to  the  excess,  crush  the  side  of  the  wood 
under  the  post,  and  proportionably  derange  the  framing,  if  not 
cause  a  total  failure.  It  will  be  observed  that  the  measure  of 


'FRAMING.  181 

tnis  resistance  is  useful  in  limiting  the  load  on  a  post  accord- 
ing to  the  kind  of  material  contained,  not  in  the  post,  but  in 
the  timber  upon  which  the  post  presses. 

293. — In  Table  II.  are  the  results  of  experiments  made  tc 
test  the  resistance  of  materials  to  flexure :  first,  the  flexure 
produced  by  compression,  the  force  acting  on  the  ends  of  the 
fibres  longitudinally;  secondly,  the  flexure  arising  from  the 
effects  of  a  cross  strain,  the  force  acting  on  the  side  of  the 
fibres  transversely,  the  beams  being  laid  on  chairs  or  rests. 
Of  white  oak,  No.  1,  there  were  eight  specimens,  of  2  by  4 
inches,  and  3£  feet  long,  seasoned  more  than  a  year  after  they 
were  prepared  for  experiment.  e  Of  the  other  kinds  of  wood 
there  were  from  three  to  five  specimens  of  each,  of  li  by  2$ 
inches,  and  from  .11  to  2f  feet  long.  Of  the  cast  iron  there 
were  six  specimens,  of  1  inch  square  and  1  foot  long;  and 
of  the  wrought  iron  there  were  five  specimens  of  American, 
three  of  f  by  2  inches,  and  two  of  1£  inches  square,  and  three 
specimens  of  common  English,  $  by  2  inches  ;  the  eight  speci- 
mens being  each  19  inches  long,  clear  bearing.  In  each 
case  the  result  is  the  average  of  the  stiffness  of  the  several 
specimens.  The  numbers  contained  in  the  second  column  are 
the  weights  producing  the  first  degree  of  flexure  in  a  post  or 
strut,  where  the  post  or  strut  is  one  foot  long  and  one  inch 
square ;  so,  likewise,  the  numbers  in  the  fifth  column,  and 
which  are  represented  in  the  rules  by  jE7,  are  the  weights 
required  to  deflect  a  beam  one*  inch,  where  the  beam  is  one 
foot  long,  clear  bearing,  and  one  inch  square. — (See  remarks 
upon  this,  Art.  (321.)  The  numbers  in  the  third  column  are 
equal  to  one-half  of  those  in  the  second.  The  numbers  con- 
tained in  the  fourth  column,  and  represented  by  n  in  the 
rules,  show  the  greatest  rate  of  deflection  that  the  material 
may  be  subjected  to  without  injury.  This  rate  multiplied  by 
the  length  in  feet,  equals  the  total  deflection  within  the  limits 
of  elasticity. 


182 


AMERICAN   HOUSE-CARPENTER. 


TABLE  H. FLEXURE. 


Under 

Under 

Compression. 

Cross  Strain. 

Kind  of  Material 

Specific 
Gravity. 

Pounds  pro- 

Value of 

Value  of 

Value  af 

ducing  the 
first  degree 

B  in 
the  Kules. 

the  Kules. 

Eir 
heBnles 

of  flexure. 

Hemlock  

0-4:02 

2640 

1320 

0-08794 

1240 

Spruce 

'432 

4190 

2095 

0-Q9197 

1  550 

•407 

2350 

1175 

0-1022 

1750 

Ohio  yellow  pine, 

•586 

6000 

3000 

0-049 

1970 

Chestnut,  . 

•52 

7720 

3860 

0-07541 

2330 

White  oak,  No.  1, 

•82 

0-09152 

2520 

White  oak,  No.  2, 

•805 

6950 

3475 

0-0567 

2590 

Georgia  pine,   . 

•755 

9660 

4830 

0-07723 

2970 

Locust, 

•863- 

10920 

5460 

0-06615 

3280 

Cast  iron, 

7-042 

0-0148 

30500 

Wrought  iron,  common  English, 
Wrought  iron,  American,  . 

7*576 
7-576 

0-03717 
0-04038 

45500 
51400 

PRACTICAL   RULES   FOR  COMPRESSION. 

First  Case. 

294. — To  find  the  weight  that  can  be  safely  sustained  by  a 
post,  when  the  height  of  the  post  is  less  than  ten  times  the 
diameter  if  round,  or  ten  times  the  thickness  if  rectangular, 
and  the  direction  of  the  pressure  coinciding  with  the  axis. 

Rule  I. — Multiply  the  area  of  the  cross-section  of  the  post, 
in  inches,  by  the  value  of  C  in  Table  I.,  the  product  will  be 
the  required  weight  in  pouncls. 

AC=w.  (!'.) 

Example. — A  Georgia  pine  post  is  6  feet  high,  and  in  cross- 
section,  8  x  12  inches,  what  weight  will  it  safely  sustain  ? 
The  area  =  8  x  12  =  96  inches ;  this  multiplied  by  1700,  the 
value  of  0,  in  the  table,  set  opposite  Georgia  pine,  the  result, 
163,200,  is  the  weight  in  pounds  required.  It  will  be  observed 
that  the  weight  would  be  the  same  for  a  Georgia  pine  post  of 
any  height  less  than  10  times  8  inches  =  80  inches  —  6  feet  8 


FRAMING.  183 

inches,  provided  its  breadth  and  thickness  remain  the  same, 
12  and  8  inches. 

295. — To  find  the  area  of  the  cross-section  of  a  post  to  sus- 
tain a  given 'weight  safely,  the  height  of  the  post  being  less 
than  ten  times  the  diameter  if  round,  or  ten  times  the  least 
side  if  rectangular ;  the  pressure  coinciding  with  the  axis. 

Rule  II. — Divide  the  given  weight  in  pounds  by  the  value  of 
{?,  in  Table  I.,  and  the  product  will  be  the  required  area  in  inches 

5-4  w  ' 

Example. — A  weight  of  38,400  pounds  is  to  be  sustained  by 
a  white  pine  post  4  feet  high,  what  must  be  its  area  of  section 
in  order  to  sustain  the  weight  safely  ?  Here,  38,400  divided 
by  1200,  the  value  of  C,  in  Table  I.,  set  opposite  white  pine, 
gives  a  quotient  of  32  ;  this,  therefore,  is  the  required  area, 
and  such  a  post  may  be  5  x  6'4  inches.  To  find  the  least  side, 
so  that  it  shall  not  be  less  than  one-tenth  of  the  height,  divide 
the  height,  reduced  to  inches,  by  10,  and  m-ake  the  least  side 
to  exceed  this  quotient.  The  area,  divided  by  the  least  side 
so  determined,  will  give  the  wide  side.  If,  however,  by  this 
process,  the  first  side  found  should  prove  to  be  the  greatest, 
then  the  size,  of  the  post  is  to  be  found  by  Rule  VII.,  VIIL,  or 
IX. 

296. — If  the  post  is  to  be  round,  by  reference  to  the  Table 
of  Circles  in  the  Appendix,  the  diameter  will  be  found  in  the 
column  of  diameters,  set  opposite  to  the  area  of  the  post  found 
in  the  column  of  areas,  or  opposite  to  the  next  nearest  area. 
For  example,  suppose  the  required  area,  as  just  found  by  the 
example  under  Rule  II.,  is  32  ;  by  reference  to  the  column  of 
areas,  33-1^3  is  the  nearest  to  32,  and  the  diameter  set  opposite 
is  6-5.  The  post  may,  therefore,  be  6£  inches  diameter. 

Second  Case. 
297.-  -To  ascertain  the  weight  that  can  be  sustained  safely 


184:  AMERICAN   HOUSE-CARPENTER. 

by  a  post  whose  height  is,  at  least,  ten  times  its  least  side  if 
rectangular,  or  ten  times  its  diameter  if  round,  the  direction 
of  the  pressure  coinciding  with  the  axis. 

Rule  III.  —  When  ike  post  is  round  the  weight  may  be 
found  by  this  rule  :  Multiply  the  square  of  the  diameter  in 
inches  by  the  square  of  the  diameter  in  inches,  and  multiply 
the  product  by  0*589  times  the  value  of  B,  in  Table  II.,  divide 
this  product  by  the  square  of  the  height  in  feet,  and  the  quo- 
tient will  be  the  required  weight  in  pounds. 

0-589  BJj^iy     0-589  B  I? 


Example.  —  "What  weight  will  a  Georgia  pine  post  sustain 
safely,  whose  diameter  is  10  inches  and  height  10  feet  ?  The 
square  of  the  diameter  is  100  ;  100  x  100  =  10,000.  And 
10,000  by  0-589  times  4830,  the  value  of  B,  Table  II.,  set 
opposite  Georgia  pine,  =  28,448,700,  and  this  divided  by  100, 
the  square  of  the  height,  equals  284,487,  the  weight  required, 
in  pounds. 

Rule  IY.  —  If  the  post  he  rectangular  the  weight  is  found 
by  this  rule  :  Multiply  the  area  of  the  cross-section  of  the  post 
by  the  square  of  the  thickness,  both  in  inches,  and  by  the 
value  of  J?,  Table  II.  Divide  the  product  by  the  square  of 
the  height  in  feet,  and  the  quotient  will  be  the  required 
weight  in  pounds. 

A  f  B      h  t3  B 
w  =  —  •'—  =  -#-  W 

Example.  —  "What  weight  will  a  white  pine  post  sustain 
safely,  whose  height  is  12  feet,  and  sides  8  and  12  inches  re- 
spectively ?  The  area  =  8  x  12  =  96  inches  ;  the  square  of 
the  thickness,  8,  =  64.  The  area  by  the  square  of  •the  thick- 
ness, 96  x  64,  =  6144  ;  and  this  by  1175,  the  value  of  B,  for 
white  pine,  equals  7,219,200.  This,  divided  by  144,  the 
square  of  the  he'ght  =  50,133|,  the  required  weight  in 
pounds, 


TEAMING. 


135 


V.—If  the  post  le  square,  the  weight  is  found  by  this 
rule  :  Multiply  the  value  of  B,  Table  II.,  by  the  square  of  the 
area  of  the  post  in  inches,  and  divide  the  product  by  the 
square  of  the  height  in  feet,  and  the  quotient  will  be  the 
required  weight  in  pounds. 

A9  B     &  B 

w  =  -jr  =  -iT'         >} 

Example. — "What  weight  will  a  white  oak  post  sustain 
safely,  whose  height  is  9  feet,  and  sides  each  6  inches  ?  The 
value  of  B,  set  opposite  white  oak,  is  3475  ;  this,  by  (36  x  36 
=)  1296,  the  square  of  the  area,  equals  4,503,600.  This  pro- 
duct, divided  by  81,  the  square  of  the  height,  gives  for  quo- 
tient, 55,600,  the  required  weight  in  pounds. 

298.— To  ascertain  the  size  of  a  post  to  sustain  safely  a  given 
weight  when  the  height  of  the  post  is  at  least  ten  times  the 
least  side  or  diameter. 

Rule  VI. —  When  the  post  is  to  le  round  or  cylindrical,  the 
size  may  be  obtained  by  this  rule  :  Divide  the  weight  in 
pounds  by  0*589  times  the  value  of  B,  Table  II.,  and  extract 
the  square  root  of  the  product ;  multiply  the  square  root  by 
the  height  in  feet,  and  the  square  root  of  this  product  will  be 
the  diameter  of  the  post  in  inches. 


h*  w 


Example.  —  What  must  be  the  diameter  of  a  locust  post,  10 
feet  high,  to  sustain  safely  40,000  pounds?  Here  0'589  times 
5,460,  the  value  of  B  for  locust,  Table  IL,  equals  3215-9. 
The  weight,  40.000,  divided  by  3215-9,  equals  12-438.  The 
square  root  of  this,  3*5268,  multiplied  by  10,  the  height,  equals 
35-268,  and  the  square  root  of  this  is  5'9386  or  5f|  inches,  the 
required  diameter  of  the  post. 

Rule  VII.  —  If  the  post  is  to  ~be  rectangular,  the  size  may  be 
obtained  by  this  rule  :  Multiply  the  square  of  the  height  in 
24 


186  AMERICAN   HOUSE-CARPENTER. 

feet  by  tlie  weight  in  pounds,  and  divide  the  product  by  the 
value  of  B,  Table  II.  Now,  if  the  breadth  is  known,  divide 
the  quotient  by  the  breadth  in  inches,  and  the  cube  root  of 
this  quotient  will  be  the  thickness  in  inches.  But  if  the  thick- 
ness is  known,  and  the  breadth  desired,  divide,  instead,  by  the 
cube  of  the  thickness  in  inches,  and  the  quotient  will  be  the 
breadth  in  inches. 


Example.  —  What  thickness  must  a  hemlock  post  have, 
whose  breadth  is  4  inches  and  height  12  feet,  to  sustain  safely 
1,000  pounds  ?  The  square  of  the  height  equals  144  ;  this,  by 
]  ,000,  the  weight,  equals  144,000.  This,  divided  by  1,320,  the 
value  of  B  for  hemlock,  Table  II.,  equals  109-091.  This, 
divided  by  4,  the  breadth,  equals  27*273,  and  the  cube  root 
of  this  is  3*01,  a  trifle  over  3  inches,  and  this  is  the  thickness 
required. 

Another  Example.  —  What  breadth  must  a  spruce  post  have, 
whose  thickness  is  4  inches  and  height  10  feet,  to  sustain  safely 
10,000  pounds  ?  The  square  of  the  height,  100,  by  10,000,  the 
weight,  equals  1,000,000.  This,  divided  by  2095,  the  value  of 
B,  Table  II.,  for  spruce,  equals  477'09  ;  and  this,  divided  by 
64,  the  cube  of  the  thickness,  equals  7*45,  nearly  7£  inches, 
the  breadth  required. 

Rule  VIII.  —  If  the  post  is  to  be  square,  the  size  may  be 
obtained  by  this  rule.  Divide  the  weight  in  pounds  by  the 
value  of  j#,  Table  II.,  and  multiply  the  square  root  of  the  pro- 
duct by  the  height  in  feet,  and  the  square  root  of  this  product 
will  be  the  dimension  of  a  side  of  the  post  in  inches. 


Example.  —  What  dimension  must  the  side  of  a  square  post 


FRAMING.  187 

have,  whose  height  is  15  feet,  the  post  being  of  Georgia 
pine,  to  sustain  safely  50,000  pounds  ?  The  weight  50,000, 
divided  by  4830,  the  value  of  JB,  Table  II.,  for  Georgia  pine, 
equals  10-352.  The  square  root  of  this,  3-2175,  multiplied  by 
15,  the  height,  equals  48*362,  and  the  square  root  of  this  is 
6-9472,  nearly  7  inches,  the  size  of  a  side  of  the  required 
post. 

299.  —  A  square  post  is  not  the  stiifest  that  can  be  made  from 
a  given  amount  of  material.  The  stiffest  rectangular  post  is 
that  whose  sides  are  in  proportion  as  6  is  to  10.  "When  this 
proportion  is  desired  it  may  be  obtained  by  the  following  rule. 

Rule  IX.  —  Divide  six-tenths  of  the  weight  in  pounds  by  the 
value  of  B,  Table  II.,  and  extract  the  square  root  of  the  quo- 
tient ;  multiply  the  square  root  by  the  height  in  feet,  and  then 
the  square  root  of  this  product  will  be  the  thickness  in  inches. 
The  breadth  is  equal  to  the  thickness  divided  by  0'6. 


t  = 


Example.  —  What  must  be  the  breadth  and  thickness  of  a 
white  pine  post,  10  feet  high,  to  sustain  safely  25,000  pounds. 
Here  A  of  25,000,  the  weight,  divided  by  1175,  the  value  of 
B,  Table  II.,  for  white  pine,  equals  12-766.  The  square  root 
of  this,  3-5729,  multiplied  by  10,  the  height,  equals  35*729,  and 
the  square  root  of  this  is  5-977,  nearly  6  inches,  the  thickness 
required.  Th's,  divided  by  0*6,  equals  10,  equals  the  breadth 
in  inches  required. 

300.  —  The  sides  of  a  post  may  be  obtained  in  any  desirable 
proportion  by  Rule  IX.,  simply  by  changing  the  decimal  0*6 
to  such  decimal  as  will  be  in  proportion  to  unity  as  one  side  is 
to  be  to  the  other.  For  example,  if  it  be  desired  to  have  the 
eides  in  proportion  as  10  is  to  9,  then  0'9  is  the  required 
decimal  ;  if  as  10  is  to  8,  then  0-8  is  the  decimal  ;  if  as  10 


188  AMERICAN   HOUSE-CAKPENTEK. 

is  to  7,  then  0'7  is  the  decimal  to  be  used  in  p'.ace  of  0-6 
in  the  rule.  And  generally  let  &  equal  the  broad  side  and  i 
the  narrow  side,  or  let  these  letters  represent  respectively  the 
numbers  that  the  sides  are  to  be  in  proportion  to  ;  then,  where 

*  equals  the  decimal  sought,  b  :  t  ::  1  :  x  =  —  =  the  required 

decimal,  or  fraction.  For  a  fraction  may  be  used  in  place  of 
the  decimal,  where  it  would  be  more  convenient,  as  is  the  case 
when  the  sides  are  desired  to  be  in  proportion  as  3  to  2.  Here 
3  :  2  ::  1  :  as  =  §.  This  fraction  should  be  used  in  the  rule  in 
place  of  the  decimal  0*6 — rather  than  its  equivalent  decimal ; 
simply  because  the  decimal  contains  many  figures,  and  there- 
fore would  not  be  convenient.  The  decimal  equivalent  to  f  is 
0-666666  +. 

Third  Case. 

301. — To  ascertain  what  weight  may  be  sustained  safely 
by  the  resistance  of  a  given  area  of  surface,  when  the  weight 
tends  to  split  off  the  part  pressed  against  by  causing  one  sur- 
face to  slide  on  the  other,  in  case  of  fracture. 

Rule  X. — Multiply  the  area  of  the  surface  by  the  value  of 
//,  in  Table  L,  and  the  product  will  be  the  weight  required  in 
pounds. 

AH=w.  (12.) 

Example. — The  foot  of  a  rafter  is  framed  into  the  end  of 
its  tie-beam,  so  that  the  uncut  substance  of  the  tie-beam  is  15 
inches  long  from  the  end  of  the  tie-beam  to  the  joint  of  the 
rafter ;  the  tie-beam  is  of  white  pine,  and  is  six  inches  thick  ; 
what  amount  of  horizontal  thrust  will  this  end  of  the  tie-beam 
sustain,  without  danger  of  having  the  end  of  the  tie-beam  split 
off?  Here  the  area  of  surface  that  sustains  the  pressure  is  6 
by  15  inches,  equal  to  90  inches.  This,  multiplied  by  160,  the 
value  of  H,  set  opposite  to  white  pine.  Table  L,  gives  a  product 
of  14,-iOO,  and  this  is  the  required  weight  in  pounds. 


FRAMING.  189 

302.  —  To  ascertain  the  area  of  surface  that  is  required  to 
sustain  a  given  weight  safely,  when  the  weight  tends  to  split 
off  the  part  pressed  against,  by  causing  one  surface  to  slide  on 
the  other,  in  case  of  fracture. 

Rule  XL  —  Divide  the  given  weight  in  pounds  by  the  value 
of  J7,  Table  I.,  and  the  quotient  will  be  the  required  area  in 
inches. 


Example.  —  The  load  on  a  rafter  causes  a  horizontal  thrust  at 
its.  foot  of  40,000  pounds,  tending  to  split  off  the  end  of  the  tie- 
beam,  what  must  be  the  length  of  the  tie-beam  beyond  the 
line,  where  the  foot  of  the  rafter  is  framed  into  it,  the  tie-beam 
being  of  Georgia  pine,  and  nine  inches  thick  ?  The  weight,  or 
horizontal  thrust,  40,000,  divided  by  170,  the  value  of  ZT, 
Table  L,  set  opposite  Georgia  pine,  gives  a  quotient  of  235*3. 
This,  the  area  of  surface  in  inches,  divided  by  9,  the  breadth 
of  the  surface  strained,  (equal  to  the  thickness  of  the  tie-beam,) 
the  quotient,  26'1,  is  the  length  in  inches  from  the  end  of 
the  tie-beam  to  the  rafter  joint,  say  26  inches. 

303.  —  A  knowledge  of  this  kind  of  resistance  of  materials  is 
useful,  also,  in  ascertaining  the  length  of  framed  tenons,  so  as 
to  prevent  the  pin,  or  key,  with  which  they  are  fastened  from 
tearing  out  ;    and,  also,  in    cases  where  tie-beams,  or  other 
timber  under  a  tensile  strain,  are  spliced,  this  rule  gives  the 
length  of  the  joggle  on  each  end  of  the  splice. 

Fourth  Case. 

304.  —  To  ascertain  what  weight  a  post  may  be  loaded  with, 
so  as  not  to  crush  the  surface  against  which  it  presses. 

Rule  XII.  —  Multiply  the  area  of  the  post  in  inches  by  the 
value  of  .P,  Table  I.,  and  the  product  is  the  weight  required  in 
pounds, 

w  =  A  P.  (14.) 


190  AMERICAN   HOUSE-CARPENTER. 

Example. — A  post,  8  by  10  inches,  stands  upon  a  white  fine 
girder ;  the  area  equals  8  x  10  -  80  inches.  This,  by  300,  the 
value  of  P,  Table  I.,  set  opposite  white  pine,  the  product, 
24,000,  is  the  required  weight  in  poinds. 

305. — To  ascertain  what  area  a  post  must  have  in  order  to 
prevent  the  post,  loaded  with  a  given  weight,  from  crushing 
the  surface  against  which  it  presses. 

Rule  XIII. — Divide  the  given  weight  in  pounds  by  the  value 
of  P,  Table  I.,  and  the  quotient  will  be  the  area  required  in 
inches. 

*=$•  (15.) 

Example. — A  post  standing  on  a  Georgia  pine  girder  is 
loaded  with  100,000  pounds,  what  must  be  its  area?  The 
weight,  100,000,  divided  by  850,  the  value  of  P,  Table  L,  set 
opposite  Georgia  pine,  the  quotient,  117-65,  is  the  required 
area  in  inches.  The  post  may  be  10  by  llf ,  or  10  x  12  inches, 
or,  if  square,  each  side  will  be  10'84:  inches,  or  12J  inches 
diameter,  if  round. 

II. RESISTANCE  TO   TENSION. 

306. — The  resistance  of  materials  to  the  force  of  stretching, 
as  exemplified  in  the  case  of  a  rope  from  which  a  weight  is 
suspended,  is  termed  the  resistance  to  tension.  In  fibrous 
materials,  this  force  will  be  different  in  the  same  specimen,  in 
accordance  with  the  direction  in  which  the  force  acts,  whether 
in  the  direction  of  the  length  of  the  fibres,  or  at  right  angles  to 
the  direction  of  their  length.  It  has  been  found  that,  in  hard 
woods,  the  resistance  in  the  former  direction  is  about  8  to  10 
times  what  it  is  in  the  latter;  and  in  soft  woods,  straight 
grained,  such  as  white  pine,  the  resistance  is  from  16  to  20 
times.  A  knowledge  of  the  resistance  in  the  direction  of  the 
fibres  is  the  most  useful  in  practice. 

307. — In  the  following  table,  the  experiments  recorded  were 


FRAMING. 


193 


to  test  this  resistance  in  such  woods,  also  iron,  as  are  in  common 
use.  Each  specimen  was  turned  cylindrical,  and  about  2 
inches  diameter,  and  then  the  middle  part  for  10  inches  in 
length  reduced  to  fths  of  an  inch  diameter,  at  the  middle  of 
the  reduced  part,  and  gradually  increased  toward  each  end, 
where  it  was  about  an  eighth  of  an  inch  larger  at  its  junction 
with  the  enlarged  end. 

TABLE  HI. — TENSION. 


Kind  of  Material. 

Specific 
Gravity. 

"Weight  produc- 
ing fracture  per 
square  inch. 

Value  of 
T 

In  the  Kules. 

0'751 

Pounds. 

20700 

3450 

Locust,              
Maple,      
White  pine,      
Ash  

•794 
•694 
•458 
•608 

15,900 
15,400 
.   14,200 
11,700 

2,650 
2,567 
2,867 
1,950 

Oak,          

•728 

10,000 

1,667 

"White  oak,        

•774 

17,000 

2,833 

'650 

17,000 

2  833 

c^  1™,  jf-  :    :    :    : 

American  wrought  iron,  2  in.  diam., 
Do.             do.        $•  and  -J-      do., 
Do.            do.  wire,  No.  3,  . 
Do.            do.     do.   No.  0,  . 
Do.     annealed  do.   No.  0,  . 

7-200 
7-600 
7'000 
7-800 

17,000 
30,000 
30,000 
55,000 
102,000 
74,500 
53,000 

2,833 
6,000 
6,000 
9,166 
17,000 
12,416 
8,833 

308. — The  value  of  Tin  the  rules,  as  contained  in  the  last 
column  of  the  above  table,  is  one-sixth  of  the  weight  pro- 
ducing fracture  per  square  inch  of  cross  section,  as  recorded  in 
the  preceding  column.  This  proportion  of  the  breaking 
weight  is  deemed  the  proper  one,  from  the  fact  that  in  prac- 
tice, through  defects  in  workmanship,  the  attachments  may  be 
so  made  as  to  cause  the  strain  to  act  along  one  side  of  the 
piece,  instead  of  through  its  axis ;  and  as  in  this  case  it  has 
been  found,  that  fracture  will  be  produced  with  $  of  the  strain 
that  can  be  sustained  through  the  axis,  therefore  one  half  of 
this  reduced  strain,  (equal  to  £  of  the  strain  through  the  axis), 
is  the  largest  that  a  due  regard  to  security  will  permit  to  be 


102  AMERICAN   HOUSE-CARPENTER. 

used.     And  in  soine  cases  it  may  be  deemed  advisable  to  load 
the  material  with  even  a  still  smaller  strain. 

309. — To  ascertain  the  weight  or  pressure  that  may  be  safely 
applied  to  a  beam  as  a  tensile  strain. 

Rule  XIY. — Multiply  the  area  of  the  cross  section  of  the 
beam  in  inches  by  the  value  of  T,  Table  III.,  and  the  product 
will  be  the  required  weight  in  pounds.  The  cross  section  here 
intended  is  that  taken  at  the  smallest  part  of  the  beam  or  rod. 
A  beam  is  usually  cut  with  mortices  in  framing ;  the  area  will 
probably  be  smallest  at  the  severest  cutting :  the  area  used  in 
the  rule  must  be  only  of  the  uncut  fibres. 

A  T=w.  (16.) 

Example. — A  tie-beam  of  a  roof  truss  is  of  white  pine,  and 
6x10  inches ;  the  cutting  for  the  foot  of  the  rafter  reduces 
the  uncut  area  to  40  inches :  what  amount  of  horizontal  thrust 
from  the  foot  of  the  rafter  will  this  tie-beam  safely  sustain  ? 
Here  40  times  2,367,  the  value  of  T,  equals  94,680,  the  required 
weight  in  pounds. 

310. — To  ascertain  the  sectional  area  of  a  beam  or  rod  that 
will  sustain  a  given  weight  safely,  when  applied  as  a  tensile 
strain. 

Rule  XV. — Divide  the  given  weight  in  pounds  by  the  value 
of  T,  Table  III.,  and  the  quotient  will  be  the  area  required  in 
inches :  this  will  be  the  smallest  area  of  uncut  fibres.  If  the 
piece  is  to  be  cut  for  mortices,  or  for  any  other  purpose,  then 
a  sufficient  addition  is  to  be  made  to  the  result  found  by 
the  rule. 

™  =  A.  (17.) 

Example. — A  rafter  produces  a  thrust  horizontally  of 
80,000  pounds ;  the  tie-beam  is  to  be  of  oak:  what  must  the 
area  of  the  cross  section  of  the  tie-beam  be,  in  order  to  sustain 
the  rafter  safely?  The  given  weight,  80,000,  divided  by 
1,667,  the  value  of  rl\  the  quotient,  48,  is  the  area  of  uncut 


FRAMING.  1  93 

fibres.  This  should  have  usually  one-half  of  its  amount  added 
to  it  as  an  allowance  for  cutting  ;  therefore  48  -f  24  =  T2. 
The  tie-beam  may  be  6  X  12  inches. 

311.  —  In  these  rules  nothing  has  been  said  of  an  allowance 
for  the  weight  of  the  beam  itself,  in  cases  where  the  beam  is 
placed  vertically,  and  the  weight  suspended   from  the  end. 
Usually,  in  timber,  this  is  small  in  comparison  with  the  load, 
and  may  be  neglected  ;  although  in  very  long  timbers,  and  where 
accuracy  is  decidedly  essential,  it  may  form  a  part  of  the  rule. 

312.  —  Taking  the  effect  of  the  weight  of  the  beam   into 
account,  the  relation  existing  between  the  weights  and  parts 
of  the  beam,  may  be  stated  algebraically  thus  :  — 

AT=w  +  k  (18.) 

Where  A  equals  the  area  of  the  section  of  uncut  fibres,  T 
equals  the  tabular  constant  in  the  rules,  which  is  equal  to  the 
load  that  may  be  safely  trusted  on  a  rod  of  like  material  with 
the  beam  and  one  inch  square  ;  w  equals  the  load,  and  k 
equals  the  weight  of  the  beam.  Now,  the  weight  of  the  beam 
equals  its  cubical  contents  in  feet,  multiplied  by  the  weight  of 
a  cubic  foot  of  like  material  ;  and  a  cubic  foot  of  the  material 
equals  62'5  times  its  specific  gravity,  while  the  cubical  contents 

of  the  beam  in  feet  equals  —  —  L  where  R  equals  the  sectional 

m 

area  in  inches,  and  I  equals  the  length  in  feet.     Hence  — 
*»  (19.) 


where  /"equals  the  specific  gravity.  It  will  be  observed  that 
A  equals  the  sectional  area  of  the  uncut  fibres,  while  R  equals 
the  sectional  area  of  the  entire  beam  ;  and,  where  the  excess 
of  R  over  A  may  be  stated  as  a  proportional  part  of  A,  or 
when  A  +  n  A  =  7?,  (n  being  a  decimal  in  proportion  to 
unity,  as  the  excess  of  R  over  A  is  to  A,)  or 
R  —  A 

—  2  —  =  n-    Then>  [from  (180  ]  — 

25 


194  AMERICAN"    HOUSE-CARPENTER. 

A    T  =   W   -f    k 


r=  w  +  0-434 

=AT-  0-434  (n 

=  A(T-  0-434  (w,  +  I)/  2  /)        (20.) 


When  .4  is  found,  to  find  J?,  we  have  from 


}  (22.) 

As  the  excess  of  R  over  A  decreases,  n  also  decreases,  until 
finally,  when  R  =  A,n  becomes  zero.     For  — 


and  when  A  —  R,  then 


When  n  equals  zero,  it  disappears  from  the  rules,  and  (20) 
becomes 

w  =  A(T-Q-±Z4:fZ)  (23.) 

and  (21)  becomes 


and  (22)  becomes 

^  =  ^1,  (25.) 

313.  —  These  rules  stated  in  words  at  length  are  as  fol- 
iows  :  — 

To  ascertain  the  weight  that  may  be  suspended  safely  from 
a  vertical  beam,  when  the  weight  of  the  beam  itself  is  to  be 
taken  into  account,  and  when  a  portion  of  the  fibres  are  cut  in 
framing. 

XYI.  —  From  the  sectional  area  of  the  beam,  deduct 


FKAMIXG.  195 

the  sectional  area  of  uncut  fibres,  and  divide  the  remainder  by 
the  sectional  area  of  the  uncut  fibres,  and  to  the  quotient  add 
unity;  multiply  this  sum  by  0'434  times  the  specific  gravity 
of  the  beam,  and  by  its  length  in  feet ;  substract  this  product 
from  the  value  of  T,  Table  III.,  and  the  remainder,  multiplied 
by  the  sectional  area  of  the  uncut  fibres,  will  be  the  required 
weight  in  pounds. 

w  =  A  (T-  (MSI  (n  +  I)/ 1}  (20.) 

Example. — A  white  pine  beam,  set  vertically,  5x9  inches 
and  30  feet  long,  is  so  cut  by  mortices  as  to  have  remaining 
only  5x6  inches  sectional  area  of  uncut  fibres :  what  weight 
will  such  a  beam  sustain  safely,  as  a  tensile  strain  ?  The  uncut 
fibres,  5x6  =  30,  deducted  from  the  area  of  the  beam,  5x9 
=  45,  there  remains  15.  This  remainder,  divided  by  30,  the 
area  of  the  uncut  fibres,  the  quotient  is  0'5.  This  added  to 
unity,  the  sum  is  1-5.  This,  by  0*434  times  0*458,  the  specific 
gravity  set  opposite  white  pine  in  Table  III.,  and  by  30,  the 
length  of  the  beam  in  feet,  the  product  is  8'95.  This  product, 
deducted  from  2,367,  the  value  of  Tset  opposite  white  pine  in 
Table  III.,  the  remainder  is  2,358*05.  This  remainder  multi- 
plied by  30,  the  sectional  area  of  the  uncut  fibres,  the  product, 
70,741 '5,  is  the  required  weight  in  pounds. 

314:. — When  the  beam  is  uncut  for  mortices  or  other  pur- 
poses, the  former  part  of  the  rule  is  not  needed ;  the  weight 
vail  then  be  found  by  the  following  rule. 

Rule  XVII. — Deduct  0-434:  times  the  specific  gravity  of 
the  beam,  multiplied  by  its  length  in  feet,  from  the  value 
of  T,  Table  III. ;  the  remainder,  multiplied  by  the  sectional 
area  of  the  beam  in  inches,  will  be  the  required  weight  in 
pounds. 

w  =  A  (r-O-434/Z).  (23.) 

Example. — A  Georgia  pine  beam,  set  vertically,  is  25  feet 
long  and  7x9  inches  in  sectional  area :  what  weight  will 
it  sustain  safely,  as  a  tensile  strain  ?  By  the  rule,  0-434:  times 


190  AMERICAN    HOUSE-CARPENTER. 

0-65,  the  specific  gravity  of  Georgia  pine,  as  in  Table  III.,  mul 
tiplied  by  25,  the  length  in  feet,  the  product  is  7"05.  This 
product,  deducted  from  2,833,  the  value  of  T,  Table  III.,  set 
opposite  Georgia  pine,  and  the  remainder,  2,825'95,  multiplied 
by  63,  the  sectional  area,  the  product,  178,034-85,  is  the 
required  weight  in  pounds. 

315.  —  To  ascertain  the  sectional  area  of  a  vertical  beam  that 
will  safely  sustain  a  given  -tensile  strain,  where  the  weight  of 
the  beam  itself  is  to  be  considered. 

Rule  XYIII.  —  Where  the  beam  is  cut  for  mortices  or  other 
purposes,  let  the  relative  proportion  of  the  uncut  fibres  to  those 
that  are  cut,  be  as  1  is  to  n,  (n  being  a  decimal  to  be  fixed  on 
at  pleasure.)  Then  to  the  value  of  n  add  unity,  and  multiply 
ing  the  sum  by  0'434  times  the  specific  gravity  in  Table  III., 
and  by  the  length  in  feet.  Deduct  this  product  from  the 
value  of  jT,  Table  III.,  divide  the  given  weight  in  pounds  by 
this  remainder,  and  the  quotient  will  be  the  area  of  the  uncut 
fibres  in  inches.  Add  unity  to  the  value  of  n,  as  above,  and 
multiply  the  sum  by  the  area  of  the  uncut  fibres  ;  the  product 
will  be  the  required  area  of  the  beam  in  inches. 


T  -  0-434  (^  +  !)/£, 
R  =  A  (n  +  1),  (22.) 

Example.  —  A  vertical  beam  of  white  oak.  30  feet  long,  is 
required  to  resist  effectually  a  tensile  strain  of  80,000  pounds  : 
what  must  be  its  sectional  area  ?  The  relative  proportion  of 
the  uncut  fibres  is  to  be  to  those  that  are  cut  as  1  is  to  0-4. 
To  0-4,  the  value  of  n,  add  1-  ;  the  sum  is  1'4.  This,  by  0-434 
times  '774,  the  specific  gravity  of  white  oak  in  Table  III.,  and 
by  30,  the  length,  the  product  is  14-109.  This,  deducted  from 
2,833,  the  value  of  T7  for  wjiite  oak  in  Table  HI.,  the  remainder 
is  2,818-891.  The  given  weight,  80,000,  divided  by  2,818-891. 
the  remainder,  as  above,  the  quotient,  28-38,  is  the  area  of  the 
\incut  fibres.  This  multiplied  by  the  sum  of  0-4  and  l-,(or 


FRAMING.  197 

the  value  of  n  and  unity  =  1-4,)  the  prcduct,  39*732,  is  the 
required  area  of  the  beam  in  inches. 

316.  —  When  the  fibres  are  uncut,  then  their  sectional  area 
equals  the  area  of  tl.e  beam,  and  may  be  found  by  the  follow- 
ing rule. 

Rule  XIX,  —  Deduct  0'434  times  the  specific  gravity  in 
Table  III.,  multiplied  by  the  length  in  feet,  from  the  value  of 
jT,  Table  III.,  and  divide  the  weight  in  pounds  by  the  remain- 
der. The  quotient  will  be  the  required  area  in  inches. 


Example.  —  A  vertical  beam  of  locust,  15  feet  long,  fibres  all 
uncut,  is  required  to  sustain  a  tensile  strain  equal  to  25,000 
pounds:  what  must  be  its  area?  Here  0'434  times  '794:,  the 
specific  gravity  for  locust  in  Table  IIL,  multiplied  by  15,  the 
length  in  feet,  is  5  -17.  This,  from  2,650,  the  value  of  T  for 
locust,  Table  IIL,  the  remainder  is  2,644*83.  The  given 
weight,  25,000,  divided  by  2,644-83,  the  remainder,  as  above, 
the  quotient,  9'45,  will  be  the  required  area  in  inches. 

III.  -  RESISTANCE  TO   CROSS-STRAINS. 

317.  —  A  load  placed  upon  a  beam,  laid  horizontal  or  in- 
clined, tends  to  bend  it,  and  if  the  weight  be  proportionally 
large,  to  break  it.  The  power  in  the  material  that  resists  this 
bending  or  breaking,  is  termed  the  resistance  to  cross-strains, 
or  transverse  strains.  "While  in  posts  or  struts  the  material  is 
compressed  or  shortened,  and  in  ties  and  suspending-pieces  it 
is  extended  or  lengthened  ;  in  beams  subjected  to  cross-strains 
the  material  is  both  compressed  and  extended.  (See  ArL 
254.)  When  the  beam  is  bent,  the  fibres  on  the  concave  side 
are  compressed,  while  those  on  the  convex  side  are  extended. 
The  line  where  these  two  portions  of  the  beam  meet  —  that  is, 
the  portion  compressed  and  the  portion  extended  —  the  hori- 
zontal line  of  juncture,  is  termed  the  neutral  line  or  plane.  It 


198  AMERICAN    HOUSE-CARPENTER. 

is  so  called  because  at  this  line  the  fibres  are  neither  com 
pressed  nor  extended,  and  hence  are  under  no  strain  whatever. 
The  location  of  this  line  or  plane  is  not  far  from  the  middle  of 
the  depth  of  the  beam,  when  the  strain  is  not  sufficient  to 
injure  the  elasticity  of  the  material ;  but  it  removes  towards 
the  concave  or  convex  side  of  the  beam  as  the  strain  is 
increased,  until,  at  the  period  of  rupture,  its  distance  from  the 
top  of  the  beam  is  in  proportion  to  its  distance  from  the  bot- 
tom of  the  beam  as  the  tensile  strength  of  the  material  is  to  its 
compressive  strength. 

318. — In  order  that  the  strength  of  a  beam  be  injured  as 
little  as  possible  by  the  cutting  required  in  framing,  all  mor- 
tices should  be  located  at  or  near  the  middle  of  the  depth. 
There  is  a  prevalent  idea  among  some,  who  are  aware  that 
the  upper  fibres  of  a  beam  are  compressed  when  subject  to 
cross-strains,  that  it  is  not  injurious  to  cut  these  top  fibres, 
provided  that  the  cutting  be  for  the  insertion  of  another  piece 
of  timber — as  in  the  case  of  gaining  the  ends  of  beams  into 
the  side  of  a  girder.  They  suppose  that  the  piece  filled  in 
will  as  effectually  resist  the  compression  as  the  part  removed 
would  have  done,  had  it  not  been  taken  out.  Now,  besides 
the  effect  of  shrinkage,  which  of  itself  is  quite  sufficient  to 
prevent  the  proper  resistance  to  the  strain,  there  is  the  mecha- 
nical difficulty  of  fitting  the  joints  perfectly  throughout ;  and, 
also,  a  great  loss  in  the  power  of  resistance,  as  the  material  is 
so  much  less  capable  of  resistance  when  pressed  at  right  angles 
to  the  direction  of  the  fibres,  than  when  directly  with  them,  as 
the  results  of  the  experiments  in  the  tables  show. 

319. — In  treating  upon  the  resistance  to  cross-strains,  the 
subject  is  divided  naturally  into  two  parts,  viz.  stiffness  and 
strength  :  the  former  being  the  power  to  resist  deflection  or 
bending,  and  the  latter  the  resistance  to  rupture. 

320. — Resistance  to  Deflection..  When  a  load  is  placed 
upon  a  beam  supported  at  each  end,  the  beam  bends  more  or 


FKAMING.  199 

less  ;  the  distance  that  the  beam  descends  u.ider  the  operation 
of  the  load,  measured  at  the  middle  of  its  length,  is  termed  its 
deflection.  In  an  investigation  of  the  laws  of  deflection  it  has 
been  demonstrated,  and  experiments  have  confirmed  it,  that 
while  the  elasticity  of  the  material  remains  uninjured  by  the 
pressure,  or  is  injured  in  but  a  small  degree,  the  amount  of 
deflection  is  directly  in  proportion  to  the  weight  producing  it, 
and  is  as  the  cube  of  the  length  ;  and,  in  pieces  of  rectangular 
sections,  it  is  inversely  proportional  to  the  breadth  and  the 
cube  of  the  depth  :  or,  inversely  proportional  to  the  fourth 
power  of  the  side  of  a  square  beam  or  of  the  diameter  of  a 
cylindrical  one.  Or,  when  I  equals  the  length  between  the 
supports,  w  the  weight  or  pressure,  5  the  breadth,  d  the  depth, 
and  p  the  deflection  ;  then  — 


equals  a  constant  quantity  for  beams  of  all  dimensions  made 
from  a  like  material.     Also, 


where  s  equals  a  side  of  a  square  beam  ;  and 


where  D  equals  the  diameter  of  a  cylindrical  beam.  The 
constant  here  is  less  than  in  the  case  of  the  square  and  of  the 
rectangular  beams.  It  is  as  much  less  as  the.  circular  beam  is 
less  stiff  than  a  square  beam  whose  side  is  equal  to  the  diame- 
ter of  the  cylindrical  one.  The  constant,  E,  is  therefore  mul- 
tiplied by  the  decimal  0'5S9. 

321.—  It  may  be  observed  that  Evt\  (26)  and  (27)  would  be 
equal  to  w,  in  case  the  dimensions  of  the  beam  and  the 
amount  of  deflection  were  each  made  equal  to  unity  ;  and  in 
(28)  equal  to  w  divided  by  0-589.  That  is,  when  in  (26)  the 
length  is  1,  the  breadth  1,  and  the  depth  1,  then  E  wou'.d  be 


200  AMERICAN    HOUSE-CARPENTER. 

equal  to  the  weight  that  would  depress  the  beam  from  its  ori 
ginal  line  equal  to  1.  Thus — 

Is  w  1*  x  w 

L  =  ^Tp  =  1  x  1s  X  1  =  w> 

the  dimensions  all  laken  in  inches  except  the  length,  and  this 
t.-iken  in  feet.  This  is  an  extreme  state  of  the  case,  for  in  most 
kinds  of  material  this  amount  of  depression  -would  exceed  the 
limits  of  elasticity  ;  and  hence  the  rule  would  here  fail  to  give 
the  correct  relation  among  the  dimensions  and  pressure.  For 
the  law  of  deflection  as  above  stated,  (the  deflection  being 
equal  for  equal  weights,)  is  true  only  while  the  depressions  are 
small  in  comparison  with  the  length.  Nothing  useful  is, 
therefore,  derived  from  this  position  of  the  question,  except  to 
give  an  idea  of  the  nature  of  the  quantity  represented  by  the 
constant,  E;  it  being  in  reality  a  measure  of  the  stiffness  of 
the  kind  of  material  used  in  comparing  one  material  with 
another.  Whatever  may  be  the  dimensions  of  the  beam,  E^ 
calculated  by  (26,)  will  always  be  the  same  quantity  for  the 
same  material ;  but  when  various  materials  are  used,  E  will 
vary  according  to  the  flexibility  or  stiffness  of  each  particular 
material.  For  example,  jS'will  be  much  greater  for  iron  than 
for  wood  ;  and  again,  among  the  various  kinds  of  wood,  it  will 
be  larger  for  the  stiff  woods  than  for  those  that  are  flexible. 

322. — If  the  amount  of  deflection  that  would  be  proper  in 
beams  used  in  framing  generally,  (such  as  floor  beams,  girders 
and  rafters,)  were  agreed  upon,  the  rules  would  be  shortened, 
and  the  labor  of  calculation  abridged.  Tredgold  proposed  to 
make  the  deflection  in  proportion  to  the  length  of  the  beam, 
and  the  amount  at  the  rate  of  oue*fortieth  of  an  inch  (=  0'025 
inch)  for  every  foot  of  length.  He  was  undoubtedly  right  in 
the  manner  and  probably  so  in  the  rate  ;  yet,  as  this  is  a  mat- 
ter of  opinion,  it  were  better  perhaps  to  leave  the  rate  of  de- 
flection open  for  the  decision  of  those  who  use  the  rules,  and 
tl-on  it  may  be  varied  to  suit  the  peculiarities  of  each  case 


TEAMING.  20  L 

that  may  arise.  Any  deflection  within  the  limits  of  the  elas- 
ticity of  the  material,  may  be  given  to  beams  used  for  some 
purposes,  while  others  require  to  be  restricted  to  that  amount 
of  deflection  that  shall  not  be  perceptible  to  a  casual  observer. 
Let  n  represent,  in  the  decimal  of  an  inch,  the  rate  of  deflec- 
tion per  foot  of  the  length  of  the  beam  ;  then  the  product  of  n, 
multiplied  by  the  number  of  feet  contained  in  the  length  of 
the  beam,  will  equal  the  total  deflection,  =  n  1.  Now,  if  n  I  be 
substituted  for  p  in  the  formulas,  (26,)  (27)  and  (28,)  they  will 
be  rendered  more  available  for  general  use.  For  example,  let 
this  substitution  be  made  in  (26,)  and  there  results  — 
V  w  V  w 


where  I  is  in  feet,  and  5,  d  and  n  in  inches  ;  and  for  (27)  — 


also  for  (28)— 

~ 


0-589  1}'  n  I  ~  0-589  D*  ri>  \'> 
where  the  notation  is  as  before,  with  also  s  and  D  in  inches. 
In  these  formulas,  w  represents  the  weight  in  pounds  concen- 
trated at  the  middle  of  the  length  of  the  beam.  If  the  weight, 
instead  thereof,  is  equally  distributed  over  the  length  of  the 
beam,  then,  since  f  of  it  concentrated  at  the  middle  will  de- 
flect a  beam  to  the  same  depth  that  th*  whole  does  when 
equally  distributed,  (Art.  281,)  therefore  — 


•f  w  Z" 

E  =  0-589  &  n? 

where  w  equals  the  whole  of  the  equally  distributed  load, 

,  Again,  if  the  load  is  borne  by  more  beams  than  one,  laid 

parallel  to  each  other  —  as,  for  example,  a  series  or  tier  of  flooi 

26 


202  AMERICAN    HOUSE-CARPENTER. 

beams — and  the  load  is  equally  distributed  over  the  supported 
Burl'ace  or  floor  ;  then,  if/"  represents  the  number  of  .pounds  ot 
the  load  contained  on  each  square  foot  of  the  floor,  or  the 
pounds' weight  per  foot  superficial,  and  c  represents  the  dis- 
tance in  feet  between  each  two  beams,  or  rather  the  distance 
from  their  centres,  and  I  the  length  of  the  beam  in  feet,  in  the 
clear,  between  the  supports  at  the  ends ;  then  c  I  will  equal 
the.  area  of  surface  supported  by  one  of  the  beams,  andjfo  I 
will  represent  the  load  borne  by  it,  equally  distributed  over  its 
length.  Now,  if  this  representation  of  the  load  be  substituted 
for  w  in  (32,)  (33)  and  (34)  there  results— 


_ 
n  ~  0'589  D*  n 


Practical  Joules  and  Examples. 

323. — To  ascertain  the  weight,  placed  upon  the  middle  of  a 
beam,  that  will  cause  a  given  deflection. 

Rule  XX. — Multiply  the  area  of  the  cross-section  of  the 
beam  by  the  square  of  the  depth  and  by  the  rate  of  the  deflec- 
tion, all  in  inches  ^multiply  the  product  by  the  value  of  E, 
Table  II.,  and  divide  this  product  by  the  square  of  the  length 
in  feet,  and  the  quotient  will  be  the  weight  in  pounds  required. 

Example. — What  weight  can  be  supported  upon  the  middle 
of  a  Georgia  pine  girder,  ten  feet  long,  eight  inches  broad, 
and  ten  inches  deep,  the  deflection  limited  .to  three-tenths  of 
an  inch,  or  at  the  rate  of  0'03  of  an  inch  per  foot  of  the  length  ? 
Here  the  area  equals  8  X  10  =  80  ;  the  square  of  the  depth 
equals  10  x  10  =  100  :  80  x  100  =  8,000  ;  this  by  0-03,  the 
rate  of  deflection,  the  product  is  240 ;  and  this  by  2970.  the 
value  of  ^for  Georgia  pine,  Table  II.,  equals  712,800.  This 


FRAMING.  203 

product,  divided  by  100,  the  square  of  the  length,  the  quot  ient, 
7,128,  is  the  weight  required  in  pounds. 

Rule  XXI. — Where  the  beam  is  square  the  weight  may  be 
found  by  the  preceding  rule  or  by  this  : — Multiply  the  square 
of  the  area  of  the  cross-section  by  the  rate  of  deflection,  both 
in  inches,  and  the  product  by  the  value  of  E,  Table  II.,  and 
divide  this  product  by  the  square  of  the  length  in  feet,  and  the 
quotient  will  be  the  weight  required  in  pounds. 

Example. — What  weight  placed  on  the  middle  of  a  spruce 
beam  will  deflect  it  sc.ven-tenths  of  an  inch,  the  beam  being 
20  feet  long,  6  inches  broad,  and  6  inches  deep  ?  Here  the 
area  is  6  X  6  =  36,  and  its  square  is  36  x  36  =  1296  ;  the  rate 
of  deflection  is  equal  to  the  total  deflection  divided  by  the 

length,   =  |^  =  0-035  ;  therefore,  1296  x  0-035  =  45-36,  and 

this  by  1550,  the  value  of  E  for  spruce,  Table  II.,  equals 
70,308.  This,  divided  by  400,  the  square  of  the  length,  equals 
175-77,  the  required  weight  in  pounds. 

Rule  XXII. — When  the  beam  is  round  find  the  weight  by 
this  rule  : — Multiply  the  square  of  the  diameter  of  the  cross- 
section  by  the  square  of  the  diameter,  and  the  product  by  the 
rate  of  deflection,  all  in  inches,  and  this  product  by  0*589 
times  the  value  of  E,  Table  II.  This  last  product,  divided  by 
the  square  of  the  length  in  feet,  will  give  the  required  weight 
in  pounds. 

Example. — What  weight  on  the  middle  of  a.  round  white 
pine  beam  will  cause  a  deflection  of  0*028  of  an  inch  per  foot, 
the  beam  being  10  inches  diameter  and  20  feet  long  ?  The 
square  of  the  diameter  equals  10  x  10  =  100  ;  100  x  100  = 
10,000 ;  this  by  the  rate,  0-028,  =  280,  and  this  by  0-589  x 
1750,  the  value  of  E,  Table  H,  for  white  pine,  equals  288,610. 
This  last  product,  divided  by  400,  the  square  of  the  length, 
equals  721'5,  the  required  weight  in  pounds. 

324. — To  ascertain  the  weight  that  will  produce  a  given  de 


204  AMERICAN    HOUSE-CARPENTEK. 

flection,  when  the  weight  is  equally  distributed  over  the  length 
of  the  beam. 

Rule  XXIII. — The  rules  for  this  are  the  same  as  the  three 
preceding  rules,  with  this  modification,  viz.,  instead  of  the 
square  of  the  length,  divide  by  five-eighths  of  the  square  of 
the  length. 

325. — In  a  series  or  tier  of  beams,  to  ascertain  the  weight 
per  foot,  equally  distributed  over  the  supported  surface,  that 
will  cause  a  given  deflection  in  the  beam. 

Rule  XXIY. — The  rules  for  this  are  the  same  as  Rules  XX., 
XXI.,  and  XXII.,  with  this  modification,  viz.,  instead  of  the 
square  of  the  length,  divide  by  the  product  of  the  distance 
apart  in  feet  between  each  twTo  beams,  (measured  from  the 
centres  of  their  breadths,)  multiplied  by  five-eighths  of  the 
cube  of  the  length,  and  the  quotient  will  be  the  required 
weight  in  pounds  that  may  be  placed  upon  each  superficial 
foot  of  the  floor  or  other  surface  supported  by  the  beams.  In 
this  and  all  the  other  rules,  the  weight  of  the  material  com- 
posing the  beams,  floor,  and  other  parts  of  the  constructions  is 
understood  to  be  a  part  of  the  load.  Therefore  from  the  ascer- 
tained weight  deduct  the  weight  of  the  framing,  floor,  plaster- 
ing, or  other  parts  of  the  construction,  and  the  remainder  will 
be  the  neat  load  required. 

Example. — In  a  tier  of  white  pine  beams,  4  x  12  inches,  20 
feet  long,  placed  16  inches  or  \\  feet  from  centres,  \vhat 
weight  per  foot  superficial  may  be  equally  distributed  over 
the  floor  covering  said  beams — the  rate  of  deflection  to  be  not 
more  than  0-025  of  an  inch  per  foot  of  the  length  of  the  beams. 
Proceeding  by  Rule  XX.  as  above  modified,  the  area  of  the 
cross-section,  4  x  12,  equals  48  ;  this  by  144,  the  square  of  the 
depth,  equals  6912,  and  this  by  0*025,  the  rate  of  deflection, 
equals  172-8.  Then  this  product,  multiplied  by  1750,  the 
value  of  E,  Table  II.,  for  white  pine,  equals  302,400.  The 
distance  between  the  centres  of  the  beams  is  H  feet,  the  cube 


FRAMING.  205 

of  the  length  is  8,000,  and  4  by  I  of  8,000  equals  6,6665.  The 
above  302,400,  divided  by  6,666^,  the  quotient,  45*36,  equals 
the  required  weight  in  pounds  per  foot  superficial.  The 
weight  of  beams,  floor  plank,  cross-furring,  and  plastering  oc- 
curring under  every  square  foot  of  .the  surface  of  the  floor,  is 
now  to  be  ascertained.  Of  the  timber  in  every  16  inches  by 
12  inches,  there  occurs  4  X  12  inches,  one  foot  long ;  this 
equals  one-third  of  a  cubic  foot.  Now,  by  proportion,  if  16 
inches  in  width  contains  ^  of  a  cubic  foot,  what  will  12  inches 

in  width  contain  ?     ?  *       =  n ^  =  fV  =  i  of  a  cubic  foot. 

lo  o  x  ID 

The  floor  plank  (Georgia  pine)  is  12  x  12  inches,  and  1£  inches 
thick,  equal  to  ^  of  a  cubic  foot,  equals  ^,  equals  j>«  Of  the 

J  -  J.33 

furring  strips,  1x2  inches,  placed  12  inches  from  centres, 
there  will  occur  one  of  a  foot  long  in  every  superficial  foot. 
Now,  since  in  a  cubic  foot  there  is  144  rods,  one  inch  square 
and  one  foot  long,  therefore,  this  furring  strip,  1  x  2  x  12 
inches,  equals  T|T  =  ^  of  a  cubic  foot.  The  weight  of  the 
timber  and  furring  strips,  being  of  white  pine,  may  be  esti- 
mated together  :  £  +  7\  =  ||  +  T'?  =  if  of  a  cubic  foot.  White 
pine  varies  from  23  to  30  pounds.  If  it  be  taken  at  30  pounds, 
the  beam  and  furring  together  will  weigh  30  x  jf  pounds, 
equals  T'92  pounds.  Georgia  pine  may  be  taken  at  50  pounds 
per  cubic  foot  ;*  the  weight  of  the  floor  plank,  then,  is  50  X 
?5T  =  5*21  pounds.  A  superficial  foot  of  lath  and  plastering 
will  weigh  about  10  Ibs.  Thus,  the  white  pine,  7*92,  Georgia 
pine,  5*21,  and  the  plastering,  10,  together  equal  23*13  pounds  ; 
this  from  45'36,  as  before  ascertained,  leaves  22-23,  say  22* 
pounds,  the  neat  weight  per  foot  superficial  that  may  be 
equally  distributed  over  the  floor  as  its  load. 

*  To  get  the  weight  of  wood  or  any  other  material,  multiply  its  speci6c  gravity  by  62'5.    For  8j* 
ciflc  Gravuies  see  Tables  L,  II.,  and  in.  and  the  Appendix  for  Weight  of  Materials. 


206 


AMERICAN   HOUSE-CARPENTEK. 


326. — To  ascertain  the  weight  when  the  beam  is  laid  not 
horizontal,  but  inclined. 

Rule  XXY. — In  each  of  the  foregoing  rules,  multiply  the 
result  there  obtained  by  the  length  in  feet,  and  divide  the  pro- 
duct by  the  horizontal  distance  between  the  supports  in  feet, 
and  the  quotient  will  be  the  required  weight  in  pounds. 

The  foregoing  Rules,  stated  algebraically,  are  placed  in  the 
following  table  : — 

TABLE    IV. STIFFNESS    OF   BEAMS  ;    WEIGHT. 


When  the 
beam  is  laid 

When  the  weight  is 

When  the  beam  is 

Rect- 
angular. 

Square. 

Round. 

Horizontal 
Horizontal 
Horizontal 
Inclining 
Inclining 
Inclining 

Concentrated  at  middle,  W,  in  pounds,  equals 
Squally  distributed,  «,  in  pounds,  equals 
}y  the  foot  superficial,/  in  pounds,  equals 
Concentrated  at  middle,  w,  In  pounds,  equals 
Equally  distributed,  w,  in  pounds,  equals 
By  the  foot  superficial,/  in  pounds,  equals 

(88.1 
Sn1>d* 

(39.) 
Ens* 

(40.) 
•589  En  D* 

(41.) 
Enid- 
"X  lr 

(42.) 
Ens* 

'  irir 

(48.) 
•9424  EnD* 

(44.) 

Enbd< 

(45.) 

Ens* 

~%TF 

(46.) 
•0424  EnJ>* 

(4T.) 
Enid' 

(48.) 
Ens* 
Ik 

(49.) 
•5$9  EnD* 

~m  — 

(50.) 
E  nld' 
-%Th 

(51.) 
Ens* 
~^Th 

(52.) 
•9424  EnD* 

—  ih 

(58.) 
Enbd< 
%-Jl^h 

(54) 
Ens* 
%  c  I'  h 

(55.) 
•9421  EnD\ 
c~Vh~ 

In  the  above  table,  I  equals  the  breadth,  and  d  equals  the 
depth  of  cross-section  of  beam  ; ' s  equals  the  breadth  of  a  side 
of  a  square  beam,  and  D  equals  the  diameter  of  a  round 
beam  ;  n  equals  the  rate  of  deflection  per  foot  of  the  length  ; 


FRAMING. 


207 


/>,  s,  5,  d  and  n.  all  in  inches  ;  I  equals  the  length,  c  equals 
the  distance  between  two  parallel  beams  measured  from  the 
centres  of  their  breadth  ;  h  equals  the  horizontal  distance 
between  the  supports  of  an  inclined  beam  ;  Z,  c  and  h  in  feet ; 
w  equals  the  weight  in  pounds  on  the  beam  ;  f  equals  the 
weight  upon  each  superficial  foot  of  a  floor  or  roof  supported 
by  two  or  more  beams  laid  parallel  and  at  equal  distances 
apart;  Evs,  a  constant,  the  value  of  which  is  found  in  Table 
II. ;  r  is  any  decimal,  chosen  at  pleasure,  in  proportion  to 
unity,  as  b  is  to  d,  from  which  proportion  b  equals  d  r. 

327. — To  ascertain  the  dimensions  of  the  cross-section  of  a 
beam  to  support  the  required  weight  with  a  given  deflection. 

Rule  XXVI. — Preliminary.  When  the  weight  is  concen- 
trated at  the  middle  of  the  length.  Multiply  the  weight  in 
pounds  by  the  square  of  the  length  in  feet,  and  divide  the  pro- 
duct by  the  product  of  the  rate  of  deflection  multiplied  by  the 
value  of  J57.,  Table  II. ,  and  the  quotient  equals  a  quantity  which 
may  be  represented  by  M— referred  to  in  succeeding  rules. 

££  =  JC  (56.) 

Rule  XXVII. — Preliminary.  When  the  weight  is  equally 
distributed  over  the  length.  Multiply  five-eighths  of  the  weight 
in  pounds  by  the  square  of  the  length  in  feet,  and  divide  the 
product  by  the  rate  of  deflection  multiplied  by  the  value  of  E, 
Table  II.,  and  the  quotient  equals  a  quantity  which  may  be 
represented  by  2F— referred  to  in  succeeding  rules. 

*£=*  <«•> 

Rule  XXVIII. — Preliminary.  When  the  weight  is  given 
per  foot  superficial  and  supported  by  two  or  more  beams. 
Multiply  the  distance  apart  between  two  of  the  beams,  (mea- 
sured from  the  centres  of  their  breadth,)  by  the  cube  of  the 
length,  both  in  feet,  and  multiply  the  product  by  five-eighths 
of  the  weight  per  foot  superficial ;  divide  this  product  by  the 


208  AMERICAN    HOUSE-CARPENTER. 

product  of  the  rate  of  deflection,  multiplied  by  the  value  of  E 
Table  II,  and  the  quotient  equals  a  quantity  which  may  be 
represented  by  V—  referred  to  in  succeeding  rules. 

^/  =  U'  (58>) 

Rule  XXIX. — Preliminary.  When  the  beam,  is  laid  not 
horizontal,  but  inclining.  In  Rules  XXVI.  and  XXVIL, 
instead  of  the  square  of  the  length  multiply  by  the  length,  and 
by  the  horizontal  distance  between  the  supports,  in  feet.  And 
in  Rule  XXVIII.,  instead  of  the  cube  of  the  length,  multiply 
by  the  square  of  the  length,  and  by  the  horizontal  distance 
between  the  supports,  in  feet. 

From  (56) 


From  (57) 


From  (58) 


'liule  XXX.  —  When  the  beam  is  rectangular  to  find  the 
dimensions  of  the  cross-section.  Divide  the  quantity  repre- 
sented by  M,  N  or  U,  (in  preceding  preliminary  rules,)  by 
the  breadth  in  inches,  and  the  cube  root  of  the  quotient  will 
equal  the  required  depth  in  inches.  Or,  divide  the  quantity 
represented  by  M,  N  or  U,  by  the  cube  of  the  depth  in  inches, 
and  the  quotient  will  equal  the  required  breadth  in  inches. 
Or,  again,  if  it  be  desired  to  have  the  breadth  and  depth  in 
proportion,  as  r  is  to  unity,  (where  r  equals  any  required  deci- 
mal,) divide  the  quantity  represented  by  M,  N  or  U,  by  the 
value  of  r,  and  extract  the  square  root  of  the  quotient  :  and 
the  square  root  extracted  the  second  time,  will  equal  the  depth 
in  inches.  Multiply  the  depth  thus  found  by  the  value  of  r, 
and  the  product  will  equal  the  breadth  in  inches. 


TEAMING.  209 

Example. — To  find  the  depth.  A.  beam  of  spruce,  laid  on 
supports  with  a  clear  bearing  of  20  feet,  is  required  to  support 
a  load  of  1674  pounds  at  the  middle,  and  the  deflection  not  to 
exceed  0'05  of  an  inch  per  foot ;  what  must  be  the  depth  when 
the  breadth  is  5  inches.  By  Eule  XXVI.  for  load  at  middle  : 
the  product  of  1674,  the  weight,  by  400,  the  square  of  the 
length,  equals  669,600.  The  product  of  0'05,  the  rate  of  de- 
flection, multiplied  by  1550,  the  value  of  E,  from  Table  II., 
set  opposite  spruce,  is  77'5.  The  aforesaid  product,  669,600, 
divided  by  77'5,  equals  8640,  the  value  of  M.  Then  by  Rule 
XXX.,  8640,  the  value  of  M,  divided  by  5,  the  breadth,  the 
quotient  is  1728,  and  12,  the  cube  root  of  this,  found  in  the 
table  of  the  Appendix,  equals  the  required  depth  in  inches. 

Example. — To  find  the  breadth.  Suppose  that  in  the  last 
example  it  were  required  to  have  the  depth  13  inches ;  in  that 
case  what  must  be  the  breadth  ?  The  value  of  M,  8640,  as 
just  found,  divided  by  2197,  the  cube  of  the  depth,  equals 
3*9326,  the  required  breadth — nearly  4  inches. 

Example. — To  find  both  breadth  and  depth,  and  in  a  certain 
proportion.  Suppose,  in  the  above  example,  that  neither  the 
breadth  nor  the  depth  are  given,  but  that  they  are  desired  to 
be  in  proportion  as  0-5  is  to  1*0.  Now,  having  ascertained  the 
value  of  J/,  by  Rule  XXYL,  to  be  8640,  as  above,  then,  by 
Rule  XXX.,  8640,  divided  by  0'5,  the  ratio,  gives  for  quotient 
17,280.  The  square  root  of  this  (by  the  table  in  the  Appen- 
dix,) is  131-45,  and  the  square  root  of  this  square  root  is  11'465, 
the  required  depth.  The  breadth  equals  11'465  X  0'5,  whic'h 
equals  5'7325.  The  depth  and  breadth  may  be  II.1,-  by  5f 
inches.  In  cases  where  the  load  is  equally  distributed  over 
the  length  of  the  beam,  the  process  is  precisely  the  same  as  set 
forth  in  the  three  preceding  examples,  except  that  five-eighths 
of  the  weight  is  to  be  used  in  place  of  the  whole  weight ;  ar.rl 
hence  it  would  be  a  useless  repetition  to  give  examples  to 
illustrate  such  cases. 

27 


210  AMERICAN   HOUSE-CARPENTER. 

Example. —  When  the  weight  is  per  foot  superficial  to  find 
tfvc  depth.  A  floor  is  to  be  constructed  to  support  500  pounds 
on  every  superficial  foot  of  its  surface.  The  beams  to  be  ot 
white  pine,  16  feet  long  in  the  clear  of  the  supports  or  walls, 
placed  16  inches  apart,  from  centres,  to  be  4  inchef*  thick,  and 
the  amount  of  deflection  not  objectionable  provided  it  be  within 
the  limits  of  elasticity.  Proceeding  by  Eule  XXYIIL,  the 
product  of  1-J-  feet,  (equal  to  16  inches,)  multiplied  by  4096,  the 
cube  of  the  length,  equals  5461£.  This,  multiplied  by  312-5, 
(equal  to  f  of  the  weight,)  equals  1,706,666.  The  largest  rate 
of  deflection  within  the  limits  of  the  elasticity  of  white  pine  is 
0-1022,  as  per  Table  II.  This,  multiplied  by  1750,  the  value  of 
E  for  white  pine,  Table  II.,  equals  178-85.  The  former  product, 
1,706,666,  divided  by  the  latter,  178-85,  equals  9,542-5,  the 
value  of  U.  Now,  by  Kule  XXX.,  this  value  of  £7",  9,542-5, 
divided  by  4,  the  breadth,  equals  2385-6,  the  cube  root  of 
which,  13-362,  is  the  required  depth — nearly  13f  inches. 

Example. — To  find  the  breadth.  Suppose,  in  the  last  exam- 
ple, that  the  depth  is  known  but  not  the  breadth,  and  that  the 
depth  is  to  be  13  inches.  Having  found  the  value  of  27,  as 
before,  to  be  9542-5,  then  by  Eule  XXX.,  dividing  9542-5,  the 
value  of  27,  by  2197,  the  cube  of  the  depth,  gives  a  quotient  of 
4-3434  and  this  equals  the  breadth — nearly  4f  inches. 

Example. — To  find  the  depth  and  "breadth  in  a  given  propor- 
tion. Suppose,  in  the  above  example,  that  the  breadth  and 
depth  are  both  unknown,  and  that  it  is  desired  to  have  them 
ih  proportion  as  0-7  is  to  TO.  Having  fouacl  the  value  of  27, 
as  before,  to  be  9542-5,  then  by  Rule  XXX.,  dividing  9542-5, 
the  value  of  27,  by  0'7,  the  quotient  is  13,632,  the  square  root 
of  which  is  116-75,  and  the  square  root  of  this  is  10-805,  the 
depth  in  inches.  Then  10-805,  multiplied  by  0'7,  the  product, 
7-5635,  is  the  breadth  in  inches.  The  size  may  be  7rV  by  10{f 
inches. 

328.—  Example. — In  the  case  of  inclined  leims  to  find  tht 


FRAMING.  2ii 

iepth.  A  beam  of  white  pine,  10  feet  long  in  the  clear  of  the 
bearing,  and  laid  at  such  an  inclination  that  the  horizontal 
distance  between  the  supports  is  9  feet,  is  required  to  support 
12,000  pounds  at  the  centre  of  its  length,  with  the  greatest 
allowable  deflection  within  the  limits  of  elasticity  ;  what  must 
be  its  depth  when  its  breadth  is  fixed  at  6  inches  ?  By  refer- 
ence to  Table  II.  it  is  seen  that  the  greatest  value  of  n,  within 
the  limits  of  elasticity,  is  0-1022.  By  Kule  XXVI.,  for  con- 
centrated load,  and  Kule  XXIX.,  for  inclined  beams,  12,000, 
the  weight,  multiplied  by  10,  the  length,  and  by  9,  the  hori- 
zontal distance,  equals  1,080,000.  The  product  of  01022,  the 
greatest  rate  of  deflection,  by  1750,  the  value  of  E,  Table  II., 
for  white  pine,  equals  178-85.  Dividing  1,080,000  by  178-85, 
the  quotient  is  603S'5S,  the  value  of  M.  Xow,  by  Rule 
XXX.,  for  rectangular  beams,  6038'58,  the  value  of  M,  divid- 
ed by  6.  the  breadth,  the  quotient  is  1006-43.  The  cube  root 
of  this,  10-02,  a  trifle  over  10  inches,  is  the  depth  required. 

Example. — In  case  of  inclined  Yearns  to  find  the  breadth. 
In  the  last  example  suppose  the  depth  fixed  at  12  inches ; 
then  by  Rule  XXX.,  6038-58,  the  value  of  M,  as  above  found, 
divided  by  1728,  the  cube  of  the  depth,  equals  3-4945,  or 
nearly  3|  inches — the  breadth  required. 

Example. — Again,  in  case  the  breadth  and  depth  are  to  ~be  in 
a  certain  proportion'  as,  for  example,  as  0-4  is  to  unity. 
Then  by  Rule  XXX.,  6038*58,  the  value  of  J/",  found  as  above, 
divided  by  0-4,  equals  15,096*45,  the  square  root  of  which  is 
122-87,  and  the  square  root  of  this  square  root  is  11*0843,  a 
trifle  over  11  inches — the  (Jepth  required.  Again,  11  multi-. 
plied  by  the  decimal  0-4,  (as  above,)  equals  4*4,  a  little  over 
4|  inches — the  breadth  required. 

In  the  three  preceding  examples,  the  weight  is  understood 
to  be  concentrated  at  the  middle.  If,  however,  the  weight 
bad  been  equally  distributed,  the  same  process  would  have 
been  used  to  obtain  the  dimensions  of  the  cross-section,  with 


212  AMERICAN   HOU8E-CAKPENTEK. 

only  one  exception  \  viz.  •§•  of  the  weight  instead  of  the  whole 
weight  would  have  been  used.  (See  Kule  XXYII.) 

Example. — In  case  of  inclined  beams;  the  weight  per  foot 
superficial,  and  borne  by  two  or  more  beams.  A  tier  of  spruce 
heams,  laid  with  a  clear  hearing  of  10  feet,  and  at  20  inches 
apart  from  centres,  and  laid  so  inclining  that  the  horizontal 
distance  between  bearings  is  8  feet,  are  required  to  sustain  40 
pounds  per  superficial  foot,  with  a  deflection  not  to  exceed 
0'02  inch  per  foot  of  the  length;  what  must  be  the  depth 
when  the  breadth  is  3  inches  ?  Proceeding  by  Rule  XXIX. 
for  inclined  beams,  and  b}T  Rule  XXYIIL,  If,  (=  20  inches,) 
the  distance  from  centres,  multiplied  by  100,  the  square  of  the 
length,  and  by  8,  the  horizontal  distance  between  bearings, 
equals  1,333$;  this,  by  £  x  40,  five-eighths  of  the  weight. 
equals  33,333$.  This,  divided  by  0-02  x  1550,  the  rate  of 
deflection,  by  the  value  of  E,  Table  II.,  for  spruce,  equal  to 
31,  equals  1075-27,  the  value  of  V.  Now  by  Rule  XXX.  for 
rectangular  beams,  1075'27,  divided  by  3,  the  breadth,  equals 
358-42,  the  cube  root  of  which,  7*1,  is  the  required  depth  in 
inches. 

Example. — The  same  as  the  preceding ;  but  to  find  the 
breadth,  when  the  depth  is  fixed  at  8  inches.  By  Rule  XXX., 
1075-27,  the  value  of  Z7",  divided  by  512,  the  cube  of  the 
depth,  equals  2-1 — the  breadth  required  in  inches. 

Example. — The  same  as  the  next  but  one  preceding ;  but  to 
find  the  breadth  and  depth  in  the  proportion  of  0*3  to  1*0,  or 
3  to  10.  By  Rule  XXX.,  1075-27,  the  value  of  U,  divided  by 
0*3,  the  value  of  r,  equals  3584-23.  The  square  root  of  this  is 
59-869,  and  the  square  root  of  this  square  root  is  7 '737 — the 
depth  required  in  inches.  This  7'737,  multiplied  by  0'3,  the 
value  of  r,  equals  2-3211 — the  required  breadth  in  inches. 
The  dimensions  may,  therefore,  be  2Tsg  by  7f  inches. 

Rule  XXXI. —  When  the  beam  is  square  to  find  the  side. 
Extract  the  square  root  of  the  quantity  represented  by  Jf,  N 


FRAMING.  213 

or  U,  iii  preliminary  Rules  XXYL,  XXYII.  and  XXVIIL, 
and  the  square  root  of  this  square  root  will  equal  the  side 
required. 

Example. — A  beam  of  chestnut,  having  a  clear  bearing  of  8 
feet,  is  required  to  sustain  at  the  middle  a  load  of  1500 
pounds ;  what  must  be  the  size  of  its  sides  in  order  that  the 
deflection  shall  not  exceed  0'03  inch  per  foot  of  its  length  ?  By 
Rule  XXVL,  1500,  the  load,  multiplied  by  64,  the  square  of 
the  length,  equals  96,000.  This  product  divided  by  0*03  times 
2330,  the  value  of  E,  Table  II.,  for  chestnut,  gives  a  quotient 
of  1373-4,  the  quantity  represented  by  M.  Now  by  Rule 
XXXI,  the  square  root  of  1373-4  is  37'05,  and  the  square 
root  of. this  is  6 '087.  The  beam  must,  therefore,  be  6  inches 
square.  In  this  example,  had  the  load,  instead  of  being  con- 
centrated at  the  middle,  been  equally  distributed  over  its 
length,  the  side  would  have  been  equal  to  the  side  just  found, 
multiplied  by  the  fourth  root  of  £  or  of  0'625,  equal  to  6'087 
X  0-889  =  5-4  inches.  (See  Rules  XXYII.  and  XXXI.) 

Example. — In  the  case  where  the  weight  is  per  foot  superfi- 
cial and  borne  by  two  or  more  beams.  A  floor,  the  beams  of 
which  are  of  oak,  and  placed  20  inches  or  Ig  feet  apart  from 
centres,  and  which  have  a  clear  bearing  of  20  feet,  is  required 
to  sustain  200  pounds  per  superficial  foot,  the  deflection  not  to 
exceed  0-025  inch  per  foot  of  the  length,  and  the  beam  to  be 
square.  By  Rule  XX  V  111.,  If,  the  distance  from  centres, 
multiplied  by  8000,  the  cube  of  the  length,  equals  13,333!  ; 
and  this  by  125,  (being  I  of  200  pounds,)  equals  1,666,666^. 
Dividing  this  by  0'025  times  2520,  the  value  of  E,  Table  II., 
for  oak,  the  quotient  is  26,455 — a  number  represented  by  U. 
Now  by  Rule  XXXI.,  the  square  root  of  this  number  is  162-65. 
and  the  square  root  of  this  square  root  is  12'753 — the  required 
side.  The  beam  may  be  12f  inches  square. 

Example. — Inclimd  square  learns,  load  at  middle.  A  bar 
of  cast-iron,  6  feet  long  in  the  clear  of  bearings,  and  laid 


214  AMERICAN   HOUSE-CARPENTER. 

inclining  so  that  the  horizontal  distance  between  the  bearing* 
is  5  feet,  is  required  to  sustain  at  the  middle  3000  pounds,  and 
the  deflection  not  to  exceed  O'Ol  inch  per  foot  of  its  length ; 
what  must  be  the  size  of  its  sides  ? 

By  Kule  XXYI.  for  load  at  middle,  modified  by  Kule 
XXIX.  for  inclined  beams ;  3000,  the  weight,  multiplied  by  6, 
the  length,  and  by  5,  the  horizontal  distance  between  bear- 
ings, equals  90,000.  The  rate  of  deflection,  0-01,  by  30,500, 
the  value  of  E,  Table  II.,  for  cast-iron,  equals  305  ;  and  9000 
divided  by  305,  equals  295-082,  the  value  of  M.  Now  by  Kule 
XXXI.  for  square  beams,  the  square  root  of  295-082  is  IMS, 
the  square  root  of  which  is  4-145 — the  size  of  the  side  required  ; 
a  trifle  over  44,  the  bar  may,  therefore,  be  4£  inches  square. 

Example. — Same  as  preceding,  ~but  the  weight  equally  distri- 
buted. By  Kule  XXVII.  f-  of  the  weight  is  to  be  used  instead 
of  the  weight ;  therefore  295*082,  the  value  of  M,  as  above, 
multiplied  by  I,  will  equal  184-426,  the  value  of  N.  By  Kule 
XXXI.  the  square  root  of  184*426  is  13-58,  the  square  root  of 
which  is  3*685 — the  size  of  the  side  required  ;  equal  to  nearly 
3j£  inches  square. 

Example. — Same  as  preceding  case,  but  the  weight  per  foot 
superficial,  and  sustained  by  2  or  more  bars,  placed  2  feet 
from  centres,  the  load  being  250  pounds  per  foot  superficial. 
By  Kule  XXVIIf.,  modified  by  Kule  XXIX.,  the  distance 
from  centres,  2,  multiplied  by  36,  the  square  of  the  length, 
and  by  5,  the  horizontal  distance,  equals  360.  This  by  156-25, 
five-eighths  of  the  weight,  equals  56,250.  The  rate  of  deflec- 
tion, 0-01,  by  30,500,  the  value  of  E,  Table  II.,  for  cast-iron, 
equals  305.  The  above  56,250,  divided  by  305,  equals  184-426, 
the  value  of  U.  Now  by  Kule  XXXI.  the  square  root  of 
184-426.  the  value  of  U,  is  13-58,  the  square  root  of  which  is 
3-685 — the  size  of  the  side  required.  It  will  be  observed  that 
this  result  is  precisely  like  that  in  the  last  example.  This  is  as 
it  should  be,  for  each  beam  has  to  sustain  the  weight  on  2  x  6 


FRAMING.  215 

=r  12  superficial  feet,  equal  to  12  x  250,  equal  3(00  pounds; 
and  all  the  other  conditions  are  parallel. 

Rule  XXXII. —  When  the  beam  is  round  to  find  the  diame- 
ter. Divide  the  value  of  Jf,  ^or  Z7",  found  by  Rules  XXYL, 
XXVII.  or  XXYIIL,  by  the  decimal  0-589,  and  extract  the 
square  root :  and  the  square  root  of  this  square  root  wil  I  be 
the  diameter  required. 

Example. — In  the  case  of  a  concentrated  load  at  middle  A. 
round  bar  of  American  iron,  of  5  feet  clear  bearing,  is  required 
to  sustain  800  pounds  at  the  middle,  with  a  deflection  not  to 
exceed  0'02  inch  per  foot ;  what  must  be  its  diameter  ?  By 
Rule  XXVI.  for  load  at  middle,  800,  the  weight,  multiplied 
by  25,  the  square  of  the  length,  equals  20,000.  The  rate  of 
deflection,  0-02,  by  51,400,  the  value  of  E,  Table  II.,  for  Ame- 
rican wrought  iron,  equals  1028.  The  above  20,000,  divided 
by  1028,  equals  19-4552,  the  value  of  M.  Now,  by  Rule 
XXXIL,  19-4552,  the  value  of  N,  divided  by  0-58*9  equals 
33-03,  the  square  root  of  which  is  5*747,  and  the  square  root 
of  this  is  2-397,  nearly  2'4,  the  diameter  required  in  inches, 
equal  to  2f  large. 

Example. — Same  case  as  the  preceding,  but  the  load  equally 
distributed.  By  Rule  XXVII.,  five-eighths  of  the  weight  is  to 
be  used  instead  of  the  whole  weight ;  therefore  the  above 
33-03,  multiplied  by  I,  equals  20*64375,  the  square  root  of 
which  is  4-544,  and  the  square  root  of  this  square  root  is  2-132, 
the  diameter  required  in  inches,  2£  inches  large. 

Example. —  When  the  weight  is  per  foot  superficial,  and  sus- 
tained by  two  or  more  bars  or  beams.  The  conditions  being 
the  same  as  in  the  preceding  examples,  but  the  weight,  100 
pounds  per  foot,  is  to  be  sustained  on  a  series  of  round  rods, 
placed  18  inches  apart  from  centres,  equal  1*5  feet.  By  Rule 
XXVIII.,  for  weight  per  foot  superficial,  1-5,  the  distance 
from  centres,  multiplied  by  125,  the  cube  of  the  length,  and 
by  62-5,  five-eighths  of  the  weight,  equals  11,718-75.  This 


216  AMERICAN   HOU8E-CABFENTEB. 

divided  by  1028,  the  product  of  the  rate  of  deflection  by  the 
value  of  E,  as  found  in  the  preceding  example,  equals  11-4, 
the  value  of  U.  Now  by  Rule  XXVII.,  11-4,  the  value  of  Z7, 
divided  by  0-589,  equals  19-42,  the  square  root  of  which  is 
4-407,  and  the  square  root  of  this  square  root  is  2*099,  the 
diameter  required — very  nearly  2rV  inches. 

Example. —  When  the  beam  is  round  and  laid  inclining,  the 
weight  concentrated  at  the  middle.  A  round  beam  of  white 
pine,  20  feet  long  between  bearings,  and  laid  inclining  so  that 
the  horizontal  distance  between  bearings  is  18  feet,  is  required 
to  support  1250  pounds  at  the  middle,  with  a  deflection  not  to 
exceed  0'05  inch  per  foot ;  what  must  be  its  diameter  ?  By 
Rule  XXYL  for  load  at  middle,  modified  by  Rule  XXIX.  for 
inclined  beams,  1250,  the  weight,  multiplied  by  20,  the  length, 
and  by  18,  the  horizontal  distance,  equals  450,000.  The  rate 
of  deflection,  0-05,  multiplied  by  1750,  the  value  of  E]  Table 
II.,  for  white  pine,  equals  87'5.  The  above  450,000  divided 
by  87-5,  equals  5142-86,  the  value  of  M.  Now  by  Rule 
XXXII.  for  round  beams,  5142-86,  the  value  of  M,  divided  by 
0-589,  equals  8731*5,  the  square  root  of  which  is  93*44,  and 
the  square  root  of  this  square  root  is  9-667,  the  diameter  re- 
quired— equal  to  9§  inches. 

Example. — Same  as  in  preceding  example,  but  the  weight 
equally  distributed.  By  Rule  XXYIL,  five-eighths  of  the 
weight  is  to  be  used  instead  of  the  whole  weight,  therefore 
8731-5,  the  result  in  the  last  example  just  previous  to  taking 
the  square  root,  multiplied  by  $,  equals  5457'2,  the  square  root 
of  which  is  73*87,  and  the  square  root  of  this  square  root  is 
8-59,  the  diameter  required — nearly  8f  inches. 

Example. — Same  as  in  the  next  but  one  preceding  exampU, 
but  the  weight  per  foot  superficial,  and  supported  ~by  two  or 
more  beams.  A  series  of  round  hemlock  poles  or  beams,  10 
feet  long  clear  bearing,  laid  inclining  so  as  that  the  horizontal 
distance  between  the  supports  equals  Y  feet,  and  laid  2  feet 


FRAMING.  217 

and  6  inches  apart  from  centres,  are  required  to  support  20 
pounds  per  superficial  foot  without  regard  to  the  amount  of 
deflection,  provided  that  the  elasticity  of  the  material  be  not 
injured;  what  must  be  their  diameter?  By  Rule  XXVUL 
for  weight  per  foot  superficial,  modified  by  Rule  XXIX.  for 
inclined  beams,  2'5,  the  distance  from  centres,  multiplied  by 
100,  the  square  of  the  length,  and  by  7,  the  horizontal  distance 
between  bearings,  and  by  five-eighths  of  the  weight,  12*5, 
equals  21,875.  The  greatest  value  of  n,  Table  II.,  for  hem- 
lock, 0-08794,  multiplied  by  1240,  the  value -of  E,  Table  II., 
for  hemlock,  equals  109-0456.  The  above  21,875,  divided  by 
109-0456,  equals  200*6,  the  value  of  U.  Now  by  Rule 
XXXIE.,  the  above  200-6,  divided  by  0-589,  equals  340-6.  the 
square  root  of  which  is  18*46,  and  the  square  root  of  this 
square  root  is  4-296,  the  diameter  required — equal  to  4yv 
inches  nearly. 

329. — The  greater  the  depth  of  a  beam  in  proportion  to  the 
thickness,  the  greater  the  strength.  But  when  the  difference 
between  the  depth  and  the  breadth  is  great,  the  beam  must  be 
stayed,  (as  at  Fig.  228,)  to  prevent  its  falling  over  and  break- 
ing sideways.  Their  shrinking  is  another  objection  to  deep 
beams ;  but  where  these  evils  can  be  remedied,  the  advantage 
of  increasing  the  depth  is  considerable.  The  following  rule  is, 
to  find  the  strongest  form  for  a  l>eam  out  of  a  given  quantity 
of  timber. 

Rule. — Multiply  the  length  in  feet  by  the  decimal,  0'6,  and 
divide  the  given  area  in  inches  by  the  product ;  and  the 
square  of  the  quotient  will  give  the  depth  in  inches. 

Example. — What  is  the  strongest  form  for  a  beam  whose 
given  area  of  section  is  48  inches,  and  length  of  bearing  20 
feet?  The  length  in  feet,  20,  multiplied  by  the  decimal,  0*6, 
gives  12  ;  the  given  area  in  inches,  48,  divided  by  12,  gives  a 
quotient  of  4,  the  square  of  which  is  16 — this  is  the  depth  in 
inches  ;  and  the  breadth  must  be  3  inches.  A  beam  16  inches 
28 


218 


AMERICAN   HOUSE-CARPENTER, 


by  3  would  bear  twice  as  much  as  a  square  beam  of  the  same 
a*rea  of  section  ;  which  shows  how  important  it  is  to  make 
beams  deep  and  thin.  In  many  old  buildings,  and  even  in 
new  ones,  in  country  places,  the  very  reverse  of  this  has  been 
practised ;  the  principal  beams  being  oftener  laid  on  the 
broad  side  than  on  the  narrower  one. 

The  foregoing  rules,  stated  algebraically,  are  plar.ed  in  the 
following  table. 


TABLE   V. STIFFNESS    OF   BEAMS;    DIMENSIONS. 


When 
the 
beam 
is  laid 

When  the  weight  is 

Rectangular. 

Square. 

Round. 

Value  of 
depth. 

Value  of 
breadth. 

When&=rfr, 
value  of  d. 

Value  of  a 
side. 

Value  of  th» 
diameter 

(66.) 
«/   «»• 

V;589^- 

ft 

Concentrated  at  middle 
Equally  distributed 
By  the  foot  superficial 

(62.1 
*/wV 
ET5 

(68.) 
wl- 
E^Td* 

(64.) 

v& 

(69.) 

V«!?± 
Enr 

(65.) 

V"*L 

En 

(67.) 
l/K  «  I' 
~E~nb 

(68.) 

y»  ic  ft 

~E  n  d* 

(70.) 
V«  «  l» 
En 

(71.) 

V  wft 
^^T^^j? 

(72.) 

^«/£P 

.£•716 

(73.) 

m 

(74.) 
l/Xfcl1 
V  -JUT? 

(75.) 

'^ 

(76.) 

V  /c" 

-9424^^ 

(81.) 

«/«>;A 

•5S9"T7T 

Inclining 

Concentrated  at  middle 
Equally  distributed 
By  the  foot  superficial 

(77.) 
l/wlh 
^E-nb- 

(78.) 
wlh 

(79.) 
j/wlh. 

(80.) 

*£- 

(82.) 

l/X<Dlh 

y~Enb~ 

(83.) 
%v>lh 
J5-T3T 

(84.) 

VX^i* 
Enr 

(85.) 

V«-«!* 

En 

(86.) 
«/    wZA 

v^BOrlT 

(87.) 
\f*f*»* 
Enb 

(88.) 
X/et'h 
End? 

(89.) 

yyafci*h 

Enr 

(90.) 

V/^iA 

En 

(91.) 

V  /<•  **  A 

-9424  En 

In  the  above  table,  5  equals  the  breadth,  and  d  the  depth  of 
cross-section  of  beam  ;  n  equals  the  rate  of  deflection  per  foot  of 
the  length  ;  5,  d  and  n,  all  in  inches.  Also,  I  equals  the  length, 
o  the  distance  between  two  parallel  beams  measured  from  the 


FRAMING. 


219 


centres  of  their  breadth,  and  h  equals  the  horizontal  distance 
between  the  supports  of  an  inclined  beam ;  ?,  c  and  A,  all  in 
feet.  Again,  w  equals  the  weight  on  the  beam,/  equals  the 
weight  upon  each  superficial  foot  of  a  floor  or  roof,  supported 
by  two  or  more  beams  laid  parallel  and  at  equal  distances 
apart ;  w  and  /  in  pounds.  And  r  is  any  decimal,  chosen  at 
pleasure,  in  proportion  to  unity,  as  b  is  to  d — from  which  pro- 
portion b  —  d  r.  E  is  a  constant  the  value  of  which  is  found 
in  Table  II. 

330. — To  ascertain  the  scantling  of  the  stiffest  beam  that  can 
be  cut  from  a  cylinder.  Let  dad,  (Fig.  223,)  be  the  section, 
and  e  the  centre,  of  a  given  cylinder.  Draw  the  diameter, 
ab  j  upon  a  and  5,  with  the  radius  of  the  section,  describe  the 
arcs,  d  e  and  e  c  ;  join  d  and  a,  a  and  c,  c  and  5,  and  b  and  d / 
then  the  rectangle,  d  a  c  I,  will  be  a  section  of  the  beam 
required. 


Fig.  223. 


331. — Resistance  to  Rupture. — The  resistance  to  deflect-ton 
having  been  treated  of  in  the  preceding  articles,  it  now  re- 
mains to  speak  of  the  other  branch  of  resistance  to  cross 
strains,  namely,  the  resistance  to  rupture.  AVhen  a  beam  is 
laid  horizontally  and  supported  at  each  end,  its  strength  to  resist 
a  cross  strain,  caused  by  a  weight  or  vertical  pressure  at  the 
middle  of  its  length,  is  directly  as  the  breadth  and  square  of 
the  depth  and  inversely  as  the  length.  If  the  beam  is  square, 
or  the  depth  equal  to  the  breadth,  then  the  strength  is  directlj 


220  AMERICAN   HOUSE-CARPENTER. 

as  the  cube  of  a  side  of  the  beam  and  inversely  as  the  length, 
and  if  the  beam  is  round  the  strength  is  directly  as  the  cube 
of  the  d;ameter  and  inversely  as  the  length. 

When  the  weight  is  concentrated  at  any  point  in  the  length, 
the  strength  of  the  beam  is  directly  as  the  length,  breadth,  and 
square  of  the  depth,  and  inversely  as  the  product  of  the  two 
parts  into  which  the  length  is  divided  by  the  point  at  which 
the  weight  is  located. 

When  the  beam  is  laid  not  horizontal  but  inclining,  the 
strength  is  the  same  as  in  each  case  above  stated,  and  also  in 
proportion,  inversely  as  the  cosine  of  the  angle  of  inclination 
with  the  horizon,  or,  which  is  the  same  thing,  directly  as  the 
length  and  inversely  as  the  horizontal  distance  between  the 
points  of  support. 

When  the  weight  is  equally  diffused  over  the  length  of  a 
beam,  it  will  sustain  just  twice  the  weight  that  could  be  sus- 
tained at  the  middle  of  its  length. 

A  beam  secured  at  one  end  only,  will  sustain  at  the  other 
end  just  one-quarter  of  the  weight  that  could  be  sustained  at 
its  middle  were  the  beam  supported  at  each  end. 

These  relations  between  the  strain  and  the  strength  exist  in 
all  materials.  For  any  particular  kind  of  material, 

&  =  8;  V) 

8,  representing  a  constant  quantity  for  all  materials  of  like 
strength.  The  superior  strength  of  one  kind  of  material  over 
another  is  ascertained  by  experiment ;  the  value  of  S  being 
ascertained  by  a  substitution  of  the  dimensions  of  the  piece  tried 
for  the  symbols  in  the  above  formula.  Having  thus  obtained 
the  value  of  /S,  the  formula,  by  proper  inversion,  becomes  use- 
ful in  ascertaining  the  dimensions  of  a  beam  that  will  require 
a  certain  weight  to  break  it ;  or  to  ascertain  the  weight  that 
will  be  required  to  break  a  certain  beam.  It  will  be  observed 
in  the  preceding  formula,  that  if  each  of  the  dimensions  of  the 


FRAMING. 


221 


beam  equal  unity,  then  w  =  S.  Hence,  S  is  equal  to  the 
weight  required  to  break  a  beam  one  inch  square  and  one 
foot  long.  The  values  of  8,  for  various  materials,  have  been 
ascertained  from  experiment,  and  are  here  recorded  : — 


1LA.BLE   VI. STRENGTH. 


Materials. 

Value  of  8. 

Number  of 
Experiments. 

Green    late    lass 

178 

4 

Spruce        
Hemlock    

845 

6 

7 

890 

9 

Hard  white  pine 

449 

1 

Ohio  yellow  pine 
Chestnut    .... 

454 
608 
610 

2 

2 

1 

Oak     
Locust       .... 
Oast-iron  (from  1550  to  22SO) 

574 
742 
1926 

2 
2 
29 

The  specimens  broken  were  of  various  dimensions,  from  one 
foot  long  to  three  feet,  and  from  one  inch  square  to  one  by 
three  inches.  The  cast-iron  specimens  were  of  the  various 
kinds  of  iron  used  in  this  country  in  the  mechanic  arts.  S 
may  be  taken  at  2,000  for  a  good  quality  of  cast-iron.  It  is 
usual  in  determining  the  dimensions  of  a  beam  to  suppose  it 
capable  of  sustaining  safely  one-third  of  the  breaking  weight, 
and  yet  Tredgold  asserts  that  one-fifth  of  the  breaking  weight 
will  in  time  injure  the  beam  so  as  to  give  it  a  permanent  set 
or  bend,  and  Hodgkinson  says  that  cast-iron  is  injured  by  any 
weight  however  small,  or,  in  other  words,  that  it  has  no  elastic 
power.  However  this  may  be,  experience  has  proved  cast- 
iron  quite  reliable  in  sustaining  safely  immense  weights  for 
a  long  period.  Practice  has  shown  that  beams  will  sustain 
safely  from  one-third  to  one-sixth  of  their  breaking  weight. 
If  the  load  is  bid  on  quietly,  and  is  to  remain  where  laid,  at 
rest,  beams  may  be  trusted  with  one-third  of  their  breaking 
weight,  but  if  the  load  is  moveable,  or  subject  to  vibration, 


222  AMERICAN   HOUSE-CARPENTER 

• 

one-quarter,  one-fifth,  or  even,  in  some  cases,  one-sixth  is  quite 
a  sufficient  proportion  of  the  breaking  load. 

332.  —  The  dimensions  of  beams  should  be  ascertained  only 
by  means  of  the  rules  for  the  stiffness  of  materials,  (Arts.  320. 
323,  et  scq.,)  as  these  rules  show  more  accurately  the  amount 
of  pressure  the  material  is  capable  of  sustaining  without  injury. 
Yet  owing  to  the  fact  that  the  rules  for  the  strength  of  mate- 
rials are  somewhat  shorter,  they  are  more  frequently  used 
than  those  for  the  stiffness  of  materials.  In  order  that  the 
proportion  of  the  breaking  weight  may  be  adjusted  to  suit  cir- 
cumstances it  is  well  to  introduce  into  the  formula  a  symbol 
to  represent  it.  The  proportion  represented  by  the  symbol 
may  then  bo  varied  at  discretion.  Let  this  symbol  be  #;  a 
decimal  in  proportion  to  unity  as  the  safe  load  is  to  the  break- 
ing load,  then  S  a  will  equal  the  safe  load.  Hence, 

S  a  1)  d2 
w  =  -  j  -  .  (93.) 

for  a  safe  load  at  middle  on  a  horizontal  beam  supported  at 
both  ends  ;  and 


for  a  safe  load  equally  diffused  over  the  length  of  the  beam  : 
and 


(95.) 


for  the  load,  per  superficial  foot,  that  can  be  sustained  safely 
upon  a  floor  supported  by  two  or  more  beams,  c  being  the  dis- 
tance in  feet  from  centres  between  each  two  beams,  and  /  the 
load  in  pounds  per  superficial  foot  of  the  floor.  Generally,  in 
(93,)  (94,)  and  (95,)  w  equals  the  load  in  pounds  ;  &  a  constant, 
the  value  of  which  is  found  in  Table  VI.  ;  a  a  decimal,  in  pro- 
portion to  unity  as  the  safe  load  is  to  the  breaking  load  ;  I  the 
length  in  feet  between  the  bearings  ;  and  I  and  d  the  breadth 
and  depth  in  inches. 


FRAMING. 


223 


TO   FIND   THE   WEIGE  T. 

333. — The  formulas  for  ascertaining  the  weight  in  tho  seve- 
ral cases  are  arranged  in  the  following  table,  where  <?,  /",  W,  Sr 
a,  I,  b  and  d  represent  as  above ;  and  also  s  equals  a  side  of 
a  square  beam ;  D  equals  the  diameter  of  a  cylindrical  beam  ; 
tn  and  n  equal  respectively  the  two  parts  into  which  the  length 
is  divided  by  the  point  at  which  the  weight  is  located  ;  and  h 
equals  the  horizontal  distance  between  the  supports  of  an 
inclined  beam. 

TABLE  VH. STRENGTH  OF  BEAMS',  SAFE  WEIGHT. 


Then  the 

When  the  beam  i 

am  is  laid 

Eectangular. 

Square. 

Konnd. 

Concentrated  at  middle,  w,  in 
pounds,  equals 

(96.) 
Sabd* 
I 

(97.) 
Sas* 
I 

(98.) 
•589  D*8  a 

1 

Squally     distributed,     v>,     in 
pounds,  equals 

(99.) 
2Sabd* 
'I 

(100.) 
2Sae* 

(101.) 
1-178  IPS  a. 

H 

By  the  foot  superficial,  f,  in 
pounds,  equals 

(102.) 
ZSabd* 

r~Tf~~ 

aoa) 

2£a*s 
cl- 

(104) 
1-178  L*  8  a 

Concentrated  at  any  point  in 

(105.) 
S  a  b  d*  I 

(106.) 
Salt* 

(107.) 
•UTD3  Sal 

\mn 

4mn 

m  n 

Concentrated  at  middle,  w,  in 

(108.) 
Sabd* 

(109.) 
Sag* 

(110.) 
•589  r>*S  a 

h 

'    h  ' 

h 

, 

Equally     distributed,     <w,     in 
pounds,  equals 

(111.) 
ZSabd* 

(112.) 
2Saj£ 

(119.) 
1-178  Z»  S  a 
h 

1 

By  the   foot   superficial,  f,  in 
pounds,  equals 

•       (U4-) 
ZSabd^ 

(115.) 
2  5  n  P 
chl~ 

(116.) 
l-lTSD'Sa 

Concentrated  at  any  point  in 
the  length,  w,  in  pounds,  equals 

(117.) 
Sabd'  P 
ihmn 

(118.) 
Sal's* 
4hmn 

(119.) 
•14T  />'  Sal* 

AMERICAN   HOU8E-CARPENTEB. 

Practical  Rules  and  Examples. 

XXXIII. — To  find  the  weight  that  may  be  supported 
safely  at  the  middle  of  a  beam  laid  horizontally.  Multiply 
the  value  of  S,  Table  VI.,  by  a  decimal  that  is  in  proportion 
to  unity  as  the  safe  weight  is  to  the  breaking  weight,  and 
divide  the  product  by  the  length  in  feet.  Then,  if  the  beam 
is  rectangular,  multiply  this  quotient  by  the  breadth  and  by 
the  square  of  the  depth,  and  the  product  will  be  the  required 
weight  in  pounds ;  or,  if  the  beam  is  square,  multiply  the  said 
quotient,  instead,  by  the  cube  of  a  side  of  the  beam  and  the 
product  will  be  the  required  weight  in  pounds ;  but,  if  the 
beam  is  round,  multiply  the  aforesaid  quotient,  instead,  by 
•589  times  the  cube  of  the  diameter,  and  the  product  will  be 
the  required  weight  in  pounds. 

Example. — "What  weight  will  a  rectangular  white  pine 
beam,  20  feet  long,  and  3  by  10  inches,  sustain  safely  at  the 
middle,  the  portion  of  the  breaking  weight  allowable  being 
0-3?  By  the  above  rule,  390,  the  value  of  8  for  white  pine, 
Table  VI.,  multiplied  by  0'3,  the  decimal  referred  to,  equals 
117,  and  this  divided  by  20,  the  length,  the  quotient  is  5'85. 
Now  the  beam  being  rectangular,  this  quotient  multiplied  by 
3  and  'by  100,  the  breadth  and  the  square  of  the  depth,  the 
product,  1755,  is  the  desired  weight  in  pounds. 

Example. — If  the  above  beam  had  been  square,  and  6  by  6 
inches,  then  the  quotient,  5'85,  multiplied  by  216,  the  cube  of 
6,  a  side,  the  product,  1263*6,  is  the  weight  required  in  pounds. 

Example. — If  the  above  beam  had  been  round,  and  6  inches 
diameter,  then  the  above  quotient,  5-85,  multiplied  by  '589 
times  216,  the  cube  of  the  diameter,  the  product,  744-26,  would 
be  the  required  weight  in  pounds. 

Rule  XXXIV. — To  find  the  weight  that  may  be  supported 
safely  when  equally  distributed  over  the  length  of  a  beam, 
laic1  horizontally.  Multiply  the  result  obtained,  by  Eule 


FRAMING.  225 

XXXIIL,  by  2,  and  the  product  will  be  the  required  weight 
in  pounds. 

Example. — In  the  example,  under  Rule  XXXIIL,  the  safe 
weight  at  middle  of  rectangular  beam  is  found  to  be  1755 
pounds.  This  multiplied  by  2,  the  product,  3510,  is  the  weight 
the  beam  will  bear  safely  if  equally  distributed  over  its  length. 

Example. — So  in  the  case  of  the  square  beam,  2527'2  pounds 
is  the  weight,  equally  distributed,  that  may  be  safely  sus- 
tained. 

Example. — And  for  the  round  beam  1488*52  is  the  required 
weight. 

Rule  XXXY. — To  ascertain  the  weight  per  superficial  foot 
that  may  be  safely  sustained  on  a  floor  resting  on  two  or  more 
beams  laid  horizontally  and  parallel.  Multiply  twice  the  value 
of  S,  Table  VI.,  by  the  decimal  that  is  in  proportion  to  unity, 
as  the  safe  weight  is  to  the  breaking  weight,  and  divide  tl  e 
product  by  the  square  of  the  length,  in  feet,  multiplied  by  the 
distance  apart,  in  feet,  between  the  beams  measured  from  their 
centres.  Now,  if  the  beams  are  rectangular,  multiply  this 
quotient  by  the  breadth  and  by  the  square  of  the  depth,  both 
in  incfcs,  and  the  product  will  be  the  required  weight  in 
pounds ;  or  if  the  beams  are  square,  multiply  said  quotient, 
instead,  by  the  cube  of  a  side  of  a  beam  and  the  product  will 
be  the  required  weight  in  pounds.  But  if  the  beams  are 
round,  multiply  the  aforesaid  quotient,  instead,  by  '589  times 
the  cube  of  the  diameter,  and  the  product  will  be  the  weight 
required  in  pounds. 

Example. — What  weight  may  be  safely  sustained  on  each 
foot  superficial  of  a  floor  resting  on  spruce  beams,  10  feet  long, 
3"by  9  inches,  placed  16  inches,  or  1J  feet,  from  centres:  the 
portion  of  the  breaking  weight  allowable  being  0-25  ?  By  the 
Rule,  690,  twice  the  value  of  spruce,  Table  VL,  multiplied  by 
0'25,  the  decimal  aforesaid,  equals  172'5.  This  product  divided 
by  100,  the  square  of  the  length,  multiplied  by  1£,  the  distance 
29 


226  AMERICAN   HOUSE-CARPENTER. 

from  centres,  equals  1*294.  Now  this  quotient  multiplied  by 
3j  the  breadth,  and  by  81,  the  square  of  the  depth,  the  product 
314*44  is  the  required  weight  in  pounds. 

Had  these  beams  been  square,  and  6  by  6  inches,  the  re- 
quired weight  would  be  279*5  pounds. 

Or,  if  round,  and  6  inches  diameter,  164*63  pounds. 

Rule  XXXYI. — To  ascertain  the  weight  that  may  be  sus- 
tained safely  on  a  beam  when  concentrated  at  any  point  of  its 
length.  Multiply  the  value  of  S,  Table  VI.,  by  the  decimal  in 
proportion  to  unity,  as  the  safe  weight  is  to  the  breaking 
weight,  and  by  the  length  in  feet,  and  divide  the  product  by 
four  times  the  product  of  the  two  parts,  in  feet,  into  which  the 
length  is  divided,  by  the  point  at  which  the  weight  is  concen- 
trated. Then,  if  the  beam  is  rectangular,  multiply  this  quo- 
tient by  the  breadth  and  by  the  cube  of  the  depth,  both  in 
inches,  and  the  product  will  be  the  required  weight  in  pounds. 
Or,  if  the  beam  is  square,  multiply  the  said  quotient,  instead, 
by  the  cube  of  a  side  of  the  beam,  and  the  product  w31  be  the 
required  weight  in  pounds.  But  if  the  beam  is  round,  multi- 
ply the  aforesaid  quotient  by  '589  times  the  cube  of  the  dia- 
meter, and  the  product  will  be  the  weight  required.  • 

Example. — What  weight  may  be  safely  supported  on  a 
Georgia  pine  beam,  5  by  12  inches,  and  20  feet  long ;  the  weight 
placed  at  5  feet  from  one  end,  and  the  proportion  of  the  break- 
ing weight  allowable  being  0*2  ?  By  the  rule,  510,  the  value  of 
S  for  Georgia  pine,  Table  VI.,  multiplied  by  0*2,  the  decimal 
referred  to,  equals  102  ;  this  by  20,  the  length,  equals  2040.; 
this  divided  by  300,  (=  4  X  5  x  15,)  or  4  times  the  product  of 
the  two  parts  into  which  the  length  is  divided  by  the  point  at 
which  the  weight  is  located,  equals  6*8.  The  beam  being  rect- 
angular, this  quotient  multiplied  by  5,  the  breadth,  and  by 
144,  the  square  of  the  depth,  equals  4896,  the  required  weight. 

A  beam,  8  inches  square,  other  conditions  being  the  same  as 
in  the  preceding  case,  would  sustain  safely  3481*6  pounds. 


FRAMING.  227 

And  a  round  beam,  8  inches  diameter,  will  sustain  safely, 
under  like  conditions,  2050*66  pounds. 

Rule  XXXYIL—  To  find  the  weight  that  may  be  safely 
sustained  on  inclined  beams.  Multiply  the  result  found  for 
horizontal  beams  in  preceding  rules,  by  the  length,  in  feet, 
and  divide  the  product  by  the  horizontal  distance  between  the 
supports,  in  feet,  and  the  quotient  will  be  the  required  weight. 

Example.  —  What  weight  may  be  safely  sustained  at  the 
middle  of  an  oak  beam,  6  X  10  inches,  and  10  feet  long,  (set 
inclining,  so  that  the  horizontal  distance  between  the  supports 
is  8  feet,)  the  portion  of  the  breaking  weight  allowable  being 
0-3  ?  The  result  for  a  horizontal  beam,  by  Bale  XXXIIL,  is 
10,332  pounds.  This,  multiplied  by  10,  the  length,  and  divid- 
ed by  8,  the  horizontal  distance,  equals  12,915  pounds,  the 
required  weight. 

TO   FIND   THE   DIMENSIONS. 

334.  —  The  following  table  exhibits,  algebraically,  rules  for 
ascertaining  the  dimensions  of  beams  required  to  support 
given  weights  ;  where  5  equals  the  breadth,  and  d  the  depth 
of  a  rectangular  beam,  in  inches  ;  I  the  length  between  sup- 
ports ;  h  the  horizontal  distance  between  the  supports  of  an 
inclined  beam,  and  c  the  distance  apart  of  two  parallel  beams, 
measured  from  the  centres  of  their  breadth,  Z,  A,  and  c,  in  feet; 
w  equals  the  weight  on  a  beam  ;  f  the  weight  on  each  super- 
ficial foot  of  a  floor  resting  on  two  or  more  parallel  beams  ;  R 
equals  a  load  on  a  beam,  and  in  and  n  the  distances,  respect- 
ively, at  which  R  is  located  from  the  two  supports  ;  also  P  is 
a  weight,  and  g  and  k  the  distances,  respectively,  at  which  P 


f,  R,  and  P)  all  in  pounds  ;  m,  n,  g,  and  &,  in  feet.  S  is  a 
constant,  the  value  of  which  is  found  in  Table  VI.  ;  a  is  a 
decimal  in  proportion  to  unity  as  the  safe  load  is  to  the  break 


228  AMERICAN   HOTJSE-CAKPENTEB. 

ing  load;  r  is  a  decimal  in  proportion  to  unity  as  J  is  to  dj 
from  which  b  =  d  r  /  a  and  r  to  be  chosen  at  discretion. 


TABLE   Vm. STRENGTH 


•When  the 

When  the 

Rectangular. 

oeam  is  laid 

weight  is 

Value  of  depth. 

Value  of  breadth. 

Concentrated      at 
middle 

(120.) 

.Jj    Wl 

SFF 

(121.) 
wl 
~8^d^ 

Equally  dlstrihnt- 
ed 

(125.) 
V_wl_ 
2  Sab 

(136.) 
wl 
T&ad* 

1 
H 

By  the  foot  super- 
ficial 

(180.) 
tflefi 

2Sal> 

(181.) 
ZSad* 

Concentrated      at 
any  point  In  the 
length 

(185.) 

A/4:  in  m  n 
*~sTbT 

(186.) 
4iD  mn 
Sad'l 

At  two   or  more 

(140.) 
^4t(Smn  +  Pgk  +  &c.~) 

(141.) 

4(Rmn  +  Pglk  +  «ftc.) 

length 

Sail 

Sa&l 

/oncentrated      at 
middle 

(145.) 
y  wh 
Sab 

(146.) 
wh 
~Sad3 

Equally   dlstribut- 

(150.) 
y    wh 

tttrs 

(151.) 
•wh 
ISad* 

! 

By  the  foot  super- 
ficial 

(165.) 
tffchl 
2Sa» 

(156.) 

fchl 
ZSad* 

Concentrated      at 
appoint  in  the 

(160.) 
j/4  h  w  m  n 

r^fF*r 

(161.) 

4  hw  mn 
'$ad*b' 

At  two   or  more 
points     in     the 

(165.) 
tfih(Rmn  +  Pffk  +  <tc.) 

(166.) 
4h(Rmn  +  Pffte  +  <tc.) 

length 

Sabt* 

S  a  d'  I* 

OF  BEAMS  :   DDIENSIONS. 


FRAMING. 


Square. 

Eonnd. 

When  6  =  dr.  value  of  a. 

Value  of  a  side. 

Value  of  the  diameter. 

(122.) 

VFaT 

(128.) 
8~a 

(124.) 

V  *> 

•589^  a 

(127.) 

(128.) 

(129.) 

t/     «^ 
Tirana 

(182.) 

(183.) 

(184.) 

V  /«»' 
1-178  8  a 

s  (187') 

"So~rT 

,    (m) 

339.) 

(142.) 

(148.) 

(144) 

£~ar  J 

ffol 

•147  Sal 

(147.) 
V^T 

3^ 

(149.) 
•589  £o 

(152.) 

«/M>A 

(153.) 

YSa 

(154) 

3  /       ^^ 

1-178  5  a 

(157.) 
ZS'af' 

058.) 

i-nasa 

V4ik? 

(163.) 
J/4  fiiomn 
V  ~Sa  H  ^ 

(164) 
•/  A  wmn 

(167.) 

(168.) 

(169.) 
yh(Rmn  +  Pglk  +  dkc.) 

<Sori; 

Sal 

•UlSalt 

230  AMERICAN   HOUSE-CARPENTEB. 

Practical  Mules  and  Examples. 

Rule  XXXVIII.—  Preliminary.  When  the  weight  'is  con 
centrated  at  the  middle.  Multiply  the  weight,  in  pounds,  by 
the  length,  in  feet,  and  divide  the  product  by  the  value  of  S, 
Table  YL,  multiplied  by  a  decimal  that  is  in  proportion  to 
unity  as  the  safe  weight  is  to  the  breaking  weignt,  and  the 
quotient  is  a  quantity  which  may  be  represented  by  J,  referred 
to  in  succeeding  rules. 

<£  =  J      .  (170.) 

Rule  XXXIX.  —  Preliminary.  When  the  weight  is  equally 
distributed.  One-half  of  the  quotient  obtained  by  the  preced- 
ing rule  is  a  quantity  which  may  be  represented  by  jB^  referred 
to  in  succeeding  rules. 


Rule  XL.  —  Preliminary.  When  the  weight  is  per  foot  su- 
perficial. Multiply  the  weight  per  foot  superficial,  in  pounds, 
by  the  square  of  the  length,  in  feet,  and  by  the  distance  apart 
from  centres  between  two  parallel  beams,  and  divide  the  pro- 
duct by  twice  the  value  of  S,  Table  VI.,  multiplied  by  a  deci- 
mal in  proportion  to  unity  as  the  safe  weight  is  to  the  break- 
ing weight,  and  the  quotient  is  a  quantity  which  may  be  re- 
presented by  _£,  referred  to  in  succeeding  rules. 


Rule  XLI.  —  Preliminary.  When  the  weight  is  concentrated 
at  any  point  in  the  length.  Multiply  the  distance,  in  feet, 
from  the  loaded  point  to  one  support,  by  the  distance,  in  feet, 
from  the  same  point  to  the  other  support,  and  by  four  times 
the  weight  in  pounds,  and  divide  the  product  by  the  value  of 
S,  Table  VI,  multiplied  by  a  decimal  in  proportion  to  unity 
as  the  safe  weight  is  to  the  breaking  weight,  and  by  the  length, 


FRAMING.  231 

in  feet ;  and  the  quotient  is  a  quantity  which  may  be  repre- 
sented by  Q,  referred  to  in  the  rules. 

iHrl=«  (173-> 

Rule  XLII. — Preliminary.  When  two  or  more  weights  are 
concentrated  at  any  points  in  the  length  of  the  learns.  Multi- 
ply each  weight  by  each  of  the  two  parts,  in  feet,  into  which 
the  length  is  divided  by  the  point  at  which  the  weight  is 
located,  and  divide  four  times  the  sum  of  these  products  by 
the  value  of  S,  Table  YL,  multiplied  by  a  decimal  in  propor- 
tion to  unity  as  the  safe  weight  is  to  the  breaking  weight,  and 
by  the  length,  in  feet,  and  the  quotient  is  a  quantity  which 
may  be  represented  by  V,  referred  to  in  the  rules. 

H+Sal*          "=  V  (m-} 

Rule  XLIII. — Preliminary.  When  the  learn  is  not  laid 
horizontal,  lut  inclining.  In  the  five  preceding  preliminary 
rules,  multiply  the  result  there  obtained  by  the  horizontal  dis- 
tance between  the  supports,  in  feet,  and  divide  the  product  by 
the  length,  in  feet,  and  the  quotient  in  each  case  is  to  be  used 
for  beams  when  inclined,  as  referred  to  in  succeeding  rules. 


TO   FIND   THE   DIMENSIONS. 

Rule  XLIV. — When  the  beam  is  rectangular.  To  find  the 
depth.  Divide  the  quantity  represented  in  preceding  rules  by 
i7,  Ji,  Z,  Q,  or  V,  by  the  breadth,  in  inches,  and  the  square 
root  of  the  quotient  will  be  the  depth  required  in  inches. 

To  find  the  breadth.  Divide  the  quantity  represented  by  J, 
K,  Z,  Q,  or  F,  by  the  square  of  the  depth,  and  the  quotient 
will  be  the  required  breadth,  in  inches. 

To  find  both  breadth  and  depth,  when  they  are  to  be  in  a 
given  proportion.  Divide  the  quantity  represented  by  J,  £, 
Z,  Q,  or  F,  by  a  decimal  in  proportion  to  unity  as  the  breadth 


'232  AMERICAN    HOUSE-CAKPENTER. 

is  to  be  to  the  depth,  and  the  cube  root  of  the  quotient  "will  be 
the  depth  in  inches.  Multiply  the  depth  by  the  aforesaid  de- 
cimal and  the  quotient  will  be  the  breadtf.  in  inches. 

Ecairvple. — A  locust  beam,  10  feet  long  in  the  clear  of  the 
supports,  is  required  to  sustain  safely  3,000  pounds  at.  the 
middle  of  its  length,  the  portion  of  the  breaking  weight  allow- 
able being  0*3  ;  what  is  the  required  breadth  and  depth  ? 
Proceeding  by  the  rule  for  weight  concentrated  at  middle, 
(Rule  XXXVIII.,)  3,000,  the  weight,  by  10,  the  length,  equals 
30,000.  The  value  of  S,  Table  VI.,  for  locust,  is  742  :  this  by 
0-3  the  decimal,  as  above,  equals  222*6  ;  the  30,000  aforesaid 
divided  by  this  222*6  equals  134*77,  equals  the  quantity  repre- 
sented by  J.  Now  to  find  the  depth  when  the  breadth  is  4 
inches,  134*77  divided  by  4,  the  breadth,  as  above  required, 
the  quotient  is  33'69,  and  the  square  root  of  this,  5*8,  is  the 
required  depth  in  inches.  But  to  find  the  breadth,  when  the 
depth  is  known,  let  the  depth  be  6  inches,  then  134'nT 
divided  by  36,  the  square  of  the  depth,  equals  3*74,  the  breadth 
required  in  inches.  Again,  to  find  both  breadth  and  depth  in 
a  given  proportion,  say,  as  0*6  is  to  1*0.  Here  134*77  divided 
by  0*6  equals  224-617,  the  cube  root  of  which  is  6*08,  the  re- 
quired depth  in  inches,  and  6*08  by  0*6  equals  3*648,  the  re- 
quired breadth  in  inches. 

Thus  it  is  seen,  in  this  example,  that  a  piece  of  locust  tim- 
ber, 10  feet  long,  having  3,000  pounds  concentrated  at  the 
middle  of  its  length,  as  T\  of  its  breaking  load,  is  required  to 
be  4  by  5[?  inches,  or  3|  by  6  inches,  or  3f  by  6£  inches.  If 
this  load  were  equally  diffused  over  the  length,  the  dimensions 
required  would  be  found  to  be  4  by  4'1,  or  1*87  by  6,  or  2*895 
by  4*825  inches,  in  the  three  cases  respectively. 

Example. — A  tier  of  chestnut  beams,  20  feet  long,  placed 
cne  foot  apart  from  centres,  is  required  to  sustain  100  pounds 
per  superficial  foot  upon  the  floor  laid  upon  them  :  this  load 
to  be  0*2  of  the  breaking  weight ;  what  is  the  required  dimen- 


FRAMING.  233 

sions  of  the  cross-section  ?  By  Rule  XL.,  the  rule  for  a  load 
per  foot  superficial,  100  by  20  x  20  and  by  1  equals  40,000. 
Twice  503,  the  value  of  S  for  chestnut,  Table  VI.,  by  0-2 
equals  201-2.  The  above  40,000  divided  by  201*2  equals  198-8, 
the  value  of  L.  Now  if  the  breadth  is  known,  and  is  3  inches, 
198'8  divided  by  3  equals  66*27,  the  square  root  of  which  is 
8 '14,  the  required  depth.  But  if  the  depth  is  known  and  is  9 
inches,  198-8  divided  by  (9  x  9  =)  81  equals  2-454  inches,  the 
required  breadth.  Again,  when  the  breadth  and  depth  are 
required  in  the  proportion  of  0*25  to  I'O,  then  198'8  divided 
by  0-25  equals  795*2,  the  cube  root  of  which  is  9*265,  the 
required  depth  in  inches,  and  9'265  by  0'25  equals  2'316,  the 
required  breadth  in  inches. 

Example. — A  cast-iron  bar,  10  feet  long,  is  required  to  sus- 
tain safely  5,000  pounds  placed  at  3  feet  from  one  end,  and 
consequently  at  7  feet  from  the  other  end,  the.  portion  of  the 
breaking  load  allowable  being  0*3  ;  what 'must  be  the  size  of 
the  cross  section  ?  By  Rule  XLL,  the  rule  for  a  concentrated 
load  at  any  point  in  the  length  of  the  beam,  3  x  7  X  4  x  5000 
=  420,000.  And  1926,  the  value  of  S  for  cast-iron,  Table  VI., 
by  0-3  and  by  10  equals  5778.  The  aforesaid  420,000  divided 
by  this,  5778,  equals  72*689,  the  value  of  Q.  Now  if  the 
breadth  is  fixed  at  1-5  then  72-689  divided  by  1-5  equals 
48'459,  the  square  root  of  which  is  6-96,  the  required  depth  in 
inches.  But  if  the  depth  is  fixed  at  6  inches,  then  72-689,  the 
value  of  Q,  divided  by  36,  the  square  of  6,  equals  2*019,  the 
required  breadth  in  inches.  Again,  if  the  breadth  and  depth 
are  required  in  the  proportion  0'2  to  I'O ;  then  Q,  72-689, 
divided  by  0*2,  equals*  363 -445,  the  cube  root  of  which  is  7'136, 
"the  required  depth  in  inches;  and  7*136  by  0'2  equals  1'427 
the  required  breadth  in  inches. 

Rule  XLV. — When  the  beam  is  square  to  find  the  breadth 
of  a  side.  The  cube  root  of  the  quantity  represented  by  e7,  K\ 
Z,  <2,  or  V,  in  preceding  rules,  is  the  breadth  of  the  side  required 
30 


234  AMERICAN   HOUSE-CARPENTER. 

Example. — A  Georgia  pine  beam,  10  feet  long,  is  required 
to  sustain,  as  O3  of  the  breaking  load,  a  weight  of  30,000 
pounds  equally  distributed  over  its  length,  and  the  beam  to  be 
square,  what  must  be  the  breadth  of  the  side  of  such  a  beam  2 
By  the  rule  for  an  equally  distributed  load,  (Rule  XXXIX.,) 
30,000  x  10  =  300,000,  arid  510  (the  value  of  £,  for  Georgia 
pine,  Table  VI.)  X  0'3  =  153.  300,000  divided  by  153  equals 
1960-784,  and  one-half  of  this  equals  980-392,  the  value  oflf. 
Now  the  cube  root  of  this  is  9*934  inches,  or  9-j-f,  the  required 
side.  Had  the  weight  been  concentrated  at  the  middle 
1960-784  would  be  the  value  of  J,  and  the  cube  root  of  this 
12*515,  or  12|-  inches,  would  be  the  size  of  a  side  of  the 
beam. 

Example. — A  square  oak  beam,  20  feet  long,  is  required  to 
sustain,  as  0-25  of  the  breaking  strength,  three  loads,  one  of 
8,000  pounds  at  5  feet  from  one  end,  one  of  7,000  pounds  at 
14  feet,  and  one  of  5,000  pounds  at  8  feet  from  one  end,  what 
must  be  the  breadth  of  a  side  of  the  beam?  The  value  of  $, 
for  oak,  Table  VI.,  is  574.  By  the  rule  for  this  case,  (Eule 
XLII.,)  8000  x  5  x  15  equals  600,000  ;  and  7000  X  14  x  6 
equals  588,000  ;  and  5000  x  8  x  12  equals  480,000.  The  sum 
of  these  products  is  1,668,000  ;  this  by  4  equals  6,672,000. 
Now  574  x  0-25  x  20  equals  2870,  and  the  6,672,000  divided 
by  the  2870  equals  2325,  the  number  represented  by  V ;  the 
cube  root  of  which  is  13*25,  the  required  size  of  a  side  of  the 
beam,  13-J  inches.  This  is  for  a  horizontal  beam.  Now  if 
this  beam  be  laid  inclining,  so  that  the  horizontal  distance 
between  the  bearings  is  15  feet,  then  to  find  the  size  by  the 
rule  for  this  case,  viz.  XLIIL,  the  abov3  number  V,  equal  to 
2325,  multiplied  by  15,  the  horizontal  distance,  equals  34,875, 
and  this  divided  by  20,  the  length,  equals  1743'75.  Now  by 
Eule  XLV.,  the  cube  root  of  this  is  12-04,  the  required  size  of 
a  side — 12  inches  full. 

Rule  XLVI. — "When  the  beam  is  round.     Divide  the  quan« 


FRAMING.  235 

tity  represented  by  e/",  E,  Z,  Q,  or  V  by  the  decimal  O5S9,  and 
the  cube  root  of  the  quotient  will  be  the  required  diameter. 

Example.— A  white  pine  beam  or  pole,  10  feet  long,  is  re- 
quired to  sustain,  as  the  O2  of  the  breaking  strength,  a  load 
of  5,000  pounds  concentrated  at  the  middle,  what  must  be  the 
diameter  ?  The  value  of  $,  for  white  pine,  Table  YL,  is  390. 
Now  by  the  rule  for  load  at  middle,  (XXXVIII.,)  5000  x  10 
=  50,000 ;  and  390  x  0-2  =  78 ;  and  50,000  -=-  78  =  641  =  J. 
By  this  rule,  641  -f-  0-589  =  1088-28,  the  cube  root  of  which, 
10-28,  is  the  required  diameter.  If  this  beam  be  inclined, 
so  that  the  horizontal  distance  between  the  supports  is  7  feet, 
then  to  find  the  diameter,  by  Eule  XLIIL,  value  of  J  as 
above,  6-il  multiplied  by  7  and  divided  by  10  equals  448'7. 
Now  by  this  rule,  448'7  -r-  0-589  =  761-796,  the  cube  root  of 
which,  9*133,  is  the  required  diameter. 

Example. — A  spruce  pole,  10  feet  long,  is  required  to  sus- 
tain, as  the  0-33  J  or  \  of  the  breaking  weight,  a  load  of  1,000 
pounds  at  3  feet  from  one  end,  what  must  be  the  diameter  ? 
The  value  of  S  for  spruce,  Table  VI.,  is  345.  By  the  rule  for 
this  case,  (Rule  XLI.,)  3  x  7  x  4  x  1000  =  84,000 ;  and  345 
x  \  x  10  =  1150  ;  and  84,000  -r- 1150  =  93-04,  the  value  of  Q. 
Now  by  this  rule,  73-04  H-  -589  =  124,  the  cube  root  of  which, 
4-9866,  is  the  required  diameter  in  inches. 

335. — Systems  of  Framing.  In  the  various  parts  of  framing 
known  as  floors,  partitions,  roofs,  bridges,  &c.,  each  has  a  spe- 
cific object ;  and,  in  all  designs  for  such  constructions,  this 
object  should  be  kept  clearly  in  view  ;  the  various  parts  being 
so  disposed  as  to  serve  the  design  with  the  least  quantity  of 
material.  The  simplest  form  is  the  best,  not  only  because  it  is 
the  most  economical,  but  for  many  other  reasons.  The  great 
number  of  joints,  in  a  complex  design,  render  the  construction 
liable  to  derangement  by  multiplied  compressions,  shrinkage, 
and,  in  consequence,  highly  increased  oblique  strains ;  by 
which  its  stability  and  durability  are  greatly  lessened. 


AMERICAN    HOUSE-CARPENTEK. 


FLOCKS. 


336. — Floors  are  most  generally  constructed  single,  that  isj 
eirnply  a  series  of  parallel  beams,  each,  spanning  the  width  oi 


Fig  224. 


the  floor,  as  seen  at  Fig.  224.     Occasionally  floors  are  con 


Fig.  225. 


stnicted  double,  as  at  Fig.  225  ;  and  sometimes  framed,  as  at 


FRAMING. 


237 


Fig.  226 ;  but  these  methods  are  seldom  practised,  inasmuch 
as  either  of  these  require  more  timber  than  the  single  floor. 
Where  lathing  and  plastering  is  attached  to  the  floor -beams  tc 
form  a  ceiling  below,  the  springing  of  the  beams,  by  custo- 
mary use,  is  liable  to  crack  the  plastering.  To  obviate  this  in 
good  dwellings,  the  double  and  framed  floors  have  been 
resorted  to,  but  more  in  former  times  than  now,  as  the  cross- 
furring  (a  series  of  narrow  strips  of  board  or  plank,  nailed 
transversely  to  the  underside  of  the  beams  to  receive  the  lath- 
ing for  the  plastering,)  serves  a  like  purpose  very  nearly  as  well. 
337. — In  single  floors  the  dimensions  of  the  beams  are  to  be 
ascertained  by  the  preceding  rules  for  the  stiffness  of  materials. 
These  rules  give  the  required  dimensions  for  the  various  kinds 
of  material  in  common  use.  The  rules'  may  be  somewhat 
abridged  for  ordinary  use,  if  some  of  the  quantities  repre- 
sented in  the  formula  be  made  constant  within  certain  limits. 
For  example,  if  the  load  per  foot  superficial,  and  the  rate  of 
deflection,  be  fixed,  then  these,  together  with  the  £ ,  and  the 


238  AMEKICAN   HOUSE-CAKPENTEB. 

constant  represented  by  L\  may  be  reduced  to  one  ccnstant. 
For  dwellings,  the  load  per  foot  may  be  taken  at  66  pounds, 
as  this  is  the  weight,  that  has  been  ascertained  by  experiment, 
to  arise  from  a  crowd  of  people  on  their  feet.  To  this  add  20 
for  the  weight  of  the  material  of  which  the  floor  is  composed, 
and  the  sum,  86,  is  the  value  of/,  or  the  weight  per  foot 
superficial  for  dwellings.  The  rate  of  deflection  allowable  for 
this  load  may  be  fixed  at  0'03  inch  per  foot  of  the  length. 
Then  (44)  transposed, 
5  foP 


becomes 

SxO-03X~^=5^ 
which,  reduced,  is 


cV  =  l>tf  (175.) 

Reducing  — —^-  for  five  of  the  most  common  woods,  and  there 

1  Q  AA 

results,  rejecting  small  decimals,  and  putting  — p-  =  ss9  x 
equal,  for 

Georgia  pine 0'6 

Oak 0-7 

White  pine I'O 

Spruce            1-15 

Hemlock 1-45 

Therefore,  the  rule  is  reduced  to  x  c  I*  =  I  <F.  And  for 
white  pine,  the  wood  most  used  for  floor  beams,  x  =  1-0,  and 
therefore  disappears  from  (he  formula,  rendering  it  still  more 
simple,  thus, 

cl'  =  ?>  fr  (1T6.) 

The  dimensions  of  beams  for  stores,  for  all  ordinary  business, 
may  also  be  calculated  by  this  modified  rule,  (175,)  for  it  will 
require  about  3£  times  the  weight  used  in  this  rule,  or  about 


FRAMING.  239 

300  pounds,  to  increase  the  deflection  to  the  limit  of  elasticity 
in  white  pine,  and  nearly  that  in  the  other  woods.  But  for 
warehouses,  taking  the  rate  of  deflection  at  its  limit,  and  fixing 
the  weight  per  foot  at  500  pounds,  including  the  weight  of  the 
material  of  which  the  floor  is  constructed,  and  letting  y  repre- 
sent the  constant,  then 

ycl3  =  l>d3  (177.) 

and  y  equals,  for 

Georgia  pine 1-85 

Oak 1-36 

White  pine 1-75 

Spruce 2-2 

Hemlock 2'85 

338. — Hence  to  find  the  dimensions  of  floor  beams  for  dwell- 
ings when  the  rate  of  deflection  is  0'03  inch  per  foot,  or  foi 
ordinary  stores  when  the  load  is  about  300  pounds  per  foot, 
and  the  deflection  caused  by  this  weight  is  within  the  limits 
of  the  elasticity  of  the  material,  we  have  the  following  rule  : 

Rule  XLYIL— Multiply  the  cube  of  the  length  by  the  dis- 
tance apart  between  the  beams,  (from  centres,)  both  in  feet, 
and  multiply  the  product  by  the  value  of  a?,  (Art.  337.)  Now 
to  find  the  breadth,  divide  this  product  by  the  cube  of  the 
depth  in  inches,  and  the  quotient  will  be  the  breadth  in  inches. 
But  if  the  depth  is  sought,  divide  the  said  product  by  the 
breadth  in  inches,  and  the  cube  root  of  the  quotient  will  be 
the  depth  in  inches  ;  or,  if  the  breadth  and  depth  are  to  be  in 
proportion,  as  r  is  to  unity,  r  representing  any  required  deci- 
mal, then  divide  the  aforesaid  product  by  the  value  of  r,  and 
extract  the  square  root  of  the  quotient,  and  the  square  root  of 
this  square  root  will  be  the  depth  required  in  inches,  and  the 
depth  multiplied  by  the  value  of  r  will  be  the  breadth  in 
inches. 

Example. — To  find  the  "breadth.  In  a  dwelling  or  ordinary 
store  what  must  be  the  breadth  of  the  beams,  when  placed  15 


240  AMERICAN    HOUSE-CAEPEXTKK. 

inches  from  centres,  to  support  a  floor  covering  a  span  of  16 
feet,  the  depth  being  11  inches,  the  beams  of  oak  ?  By  the 
rule,  4096,  the  cube  of  the  length,  bj  li,  the  distance  from 
centres,  and  by  0-7,  the  value  of  a?,  for  oak,  equals  3584.  This 
divided  by  1331,  the  cube  of  the  depth,  equals  2'69  inches,  or 
2}J-  inches,  the  required  bread th. 

Example. — To  find  the  depth.  The  conditions  being  the 
same  as  in  the  last  example,  what  must  be  the  depth  when  the 
breadth  is  3  inches.  The  product,  3584,  as  above,  divided  by 
3,  the  breadth,  equals  1194| ;  the  cube  root  of  this  is  10-61,  or 
lOf  inches  nearly. 

Example. — To  find  the  'breadth  and  depth  in  proportion, 
say,  as  0'3  to  1*0.  The  aforesaid  product,  3584,  divided  by 
0'3,  the  value  of  r,  equals  ll,946f,  the  square  root  of  which  is 
109'3,  and  the  square  root  of  this  is  10'45,  the  required  depth. 
This  multiplied  by  0'3,  the  value  of  r,  equals  3'135,  the  re- 
quired breadth,  the  beam  is  therefore  to  be  3£  by  10£  inches. 

339. — And  to  find  the  breadth  and  depth  of  the  beams  for  a 
floor  of  a  warehouse  sufficient  to  sustain  500  pounds  per  foot 
superficial,  (including  weight  of  the  material  in  the  floor,)  with 
a  deflection  not  exceeding  the  limits  of  the  elasticity  of  the 
material,  we  have  the  following  rule  : 

Rule  XLVni. — The  same  as  XLYIL,  with  the  exception 
that  the  value  of  y  (Art.  .337)  is  to  be  used  instead  of  the 
value  of  x. 

Example. — To  find  the  breadth.  The  beams  of  a  warehouse 
floor  are  to  be  of  Georgia  pine,  with  a  clear  bearing  between 
the  walls  of  15  feet,  and  placed  14  inches  from  centres,  what 
must  be  the  breadth  when  the,  depth  is  11  inches  ?  By  the 
rule,  3375,  the  cube  of  the  length,  and  l£,  the  distance  from 
centres,  and  T35,  the  value  of  y,  for  Georgia  pine,  all  multi- 
plied together,  equals  5315-625  ;  and  this  product  divided  by 
1331,  the  cube  of  the  depth,  equals  3'994,  the  required  depth, 
or  4  inches. 


TEAMING.  241 

Example.  —  To  find  the  d-epth.  The  conditions  remaining, 
as  in  last  example,  what  must  be  the  depth  when  the  breadth 
is  3  inches?  5315'625,  the  said  product,  divided  by  3,  the 
breadth,  equals  1771'875,  and  the  cube  root  of  this,  12-1,  or  12 
inches,  is  the  depth  required. 

Example.  —  To  find  the  breadth  and  depth  in  a  given  propor- 
tion, say,  0-35  to  ro.  5315-625  aforesaid,  divided  by  O35,  the 
value  of  r,  equals  15187*5,  the  square  root  of  which  is  121-8, 
and  the  square  root  of  this  square  root  is  11'04,  or  11  inches, 
the  required  depth.  And  11'04  multiplied  by  0'35,  the  value 
of  TJ  equals  3'864,  the  required  breadth  —  3£  inches. 

340.  —  It  is  sometimes  desirable,  when  the  breadth  and  depth 
of  the  beams  are  fixed,  or  when  the  beams  have  been  sawed 
and  are  now  ready  for  use,  to  know  the  distance  from  centres 
at  which  such  beams  should  be  placed,  in  order  that  the  floor 
be  sufficiently  stiff.  In  this  case,  (175,)  transposed,  and  put- 

1800   4l 
ting  x  =  —  ^-,  there  results 


This  in  words,  at  length,  is,  as  follows  : 

Rule  XLIX.—  Multiply  the  cube  of  the  depth  by  the 
breadth,  both  in  inches,  and  divide  the  product  by  the  cube 
of  the  length,  in  feet,  multiplied  by  the  value  of  a?,  for  dwell- 
ings, and  for  ordinary  stores,  or  by  y  for  warehouses  ;  and  the 
quotient  will  be  the  distance  apart  from  centres  in  feet. 

Example.  —  A  span  of  17  feet,  in  a  dwelling,  is  to  be  covered 
by  white  pine  beams.  3  X  12  inches,  at  what  distance  apart 
from  centres  must  they  be  placed  ?  By  the  rule,  1728,  the 
cube  of  the  depth,  multiplied  by  3,  the  breadth,  equals  5184. 
The  cube  of  17  is  4913,  this  by  TO,  the  value  of  x,  ror  white 
pine,  equals  4913.  The  aforesaid  5184,  divided  by  tin's,  4913, 
equals  1-055  feet,  or  1  foot  and  f  of  an  inch. 

341.  —  Where  chimneys,  fbes,  stairs,  etc.,  occur  to  interrupt 


242 


AMERICAN   HOUSE-CAEPENTER. 


Fig.  227 

the  bearing,  the  beams  are  framed  into  a  piece,  5,  (Fig.  227,) 
called  a  header.  The  beams,  a  #,  into  which  the  header  i8 
framed,  are  called  trimmers  or  carriage-beams.  These  framed 
beams  require  to  be  made  thicker  than  the  common  beams. 
The  header  must  be  strong  enough  to  sustain  one-half  of  the 
weight  that  is  sustained  uvon  the  tail  beams,  c  c,  (the  wall  at 
the  opposite  end  or  another  header  there  sustaining  the  other 
half,)  and  the  trimmers  must  each  sustain  one-half  of  the 
weight  sustained  by  the  header  in  addition  to  the  weight  it 
supports  as  a  common  beam.  It  is  usual  in  practice  to  make 
these  framed  beams  one  inch  thicker  than  the  common  beams 
for  dwellings,  and  two  inches  thicker  for  heavy  stores.  This 
practice  in  ordinary  cases  answers  very  well,  but  in  extreme 
cases  these  dimensions  are  not  proper.  Rules  applicable  gene- 
rally must  be  deduced  from  the  conditions  of  the  case — the 
load  to  be  sustained  and  the  strength  of  the  material. 

342.— For  the  header,  formula  (68,)  Table  V.,  is  applicable. 
The  weight,  represented  by  w,  is  equal  to  the  superficial  area 
of  the  floor  supported  by  the  header,  multiplied  by  the  load 
on  every  superficial  foot  of  the  floor.  This  is  equal  to  the 
length  of  the  header  multiplied  by  half  the  length  of  the  tail 
beams,  and  by  86  pounds  for  dwellings  and  ordinary  stores,  or 


FRAMING.  243 

by  500  pounds  for  warehouses.     Calling  the  length  of  the  tail 
beams,  in  feet,  <?,  formula  (68,)  becomes 


Then  if/  equals  86,  and  n  equals  0'03,  there  results 

(179.) 


This  in  words,  is,  as  follows  : 

Rule  L.  —  Multiply  900  times  the  length  of  the  tail  beams 
by  the  cube  of  the  length  of  the  header,  both  in  feet.  The 
product,  divided  by  the  cube  of  the  depth,  multiplied  by  the 
value  of  E,  Table  II.,  will  equal  the  breadth,  in  inches,  for 
dwellings  or  ordinary  stores. 

Example.  —  A  header  of  white  pine,  for  a  dwelling,  is  10  feet 
long,  and  sustains  tail  beams  20  feet  long,  its  depth  is  12 
inches,  what  must  be  its  breadth  ?  By  the  rule,  900  X  20  x 
10°  =  18,000,000.  This,  divided  by  (12s  X  IT50  =)  3,024,000, 
equals  5'95,  say  6  inches,  the  required  breadth. 

For  heavy  warehouses  the  rule  is  the  same  as  the  above, 
only  using  1550  in  the  place  of  the  900.  This  constant  may 
be  varied,  at  discretion,  to  anything  between  900  and  5000,  in 
accordance  with  the  use  to  which  the  floor  is  to  be  put. 

343.  —  In  regard  to  the  trimmer  or  carriage  beam,  formula 
(136,)  Table  Yin.,  is  applicable.  The  load  thrown  upon  the 
trimmer,  in  addition  to  its  load  as  a  common  beam,  is  equal 
to  one-half  of  the  load  on  the  header,  and  therefore,  as  has 
been  seen  in  last  article,  is  equal  to  one-half  of  the  superficial 
area  of  the  floor,  supported  by  the  tail  beams,  multiplied  by 
the  weight  per  superficial  foot  of  the  load  upon  the  floor; 
therefore,  when  the  length  of  the  header,  in  feet,  is  represented 

by^,  and  the  length  of  the  tail  beam  by  n,  w  equals  —  x  —  X/, 

2       2 


equals  \fj  n,  and  therefore  (136,)  of  Table  YHL,  becomes 
,      f  j  m  n9 
°  =  Saffl 


244  AMERICAN   HOUSE-CARPENTEB. 

equals  the  additional  thickness  to  be  given  to  a  common 
beam  when  used  as  a  trimmer,  and  for  dwellings  when  / 
equals  86  and  a  equals  0*3,  this  part  of  the  formula  reduces  to 
286^,  or,  for  simplicity,  call  it  300,  which  would  be  the  same 
as  fixing/  at  90  instead  of  86.  Then  we  have 


This,  in  words,  is  as  follojvs  : 

Rule  LI. — For  dwellings.  Multiply  300  times  the  length 
of  the  header  by  the  square  of  the  length  of  the  tail  beams, 
and  by  the  difference  in  length  of  the  trimmer  and  tail  beams, 
all  in  feet.  Divide  this  product  by  the  square  of  the  depth  in 
inches,  multiplied  by  the  length  of  the  trimmer  in  feet,  and  by 
the  value  of  S,  Table  VI,  and  the  quotient  added  to  the  thick- 
ness of  a  common  beam  of  the  floor,  will  equal  the  required 
thickness  of  the  trimmer  beam. 

Example. — A  tier  of  3  x  12  inch  beams  of  white  pine,  hav- 
ing a  clear  bearing  of  20  feet,  has  a  framed  well-hole  at  one 
side,  of  5  by  12  feet,  the  header  being  12  feet  long,  what  must 
be  the  thickness  of  the  trimmer  beams  ?  By  the  rule,  300  x 
12  x  152  x  5,  divided  by  the  product  of  12*  x  20  x  390,  equals 
3*6,  and  this  added  to  3,  the  thickness  of  one  of  the  common 
beams,  equals  6'6,  the  breadth  required,  6£  inches. 

For  stores  and  warehouses  the  rule  is  the  same  as  the  above, 
only  the  constant,  300,  must  be  enlarged  in  proportion  to  the 
load  intended  for  the  floor,  making  it  as  high  as  1600  for 
heavy  warehouses. 

344. — "When  a  framed  opening  occurs  at  any  point  removed 
from  the  wall,  requiring  two  headers,  then  the  load  from  the 
headers  rest  at  two  points  on  the  carriage  beam,  and  here  for- 
mula (141.)  Table  YIIL,  is  applicable.  In  this  special,  case 
this  formula  reduces  to 


FRAMING.  245 

where  5  equals  the  additional  thickness,  in  inches,  to  be  given 
to  the  carriage  beam  over  the  thickness  of  the  common  beams ; 
j,  the  length  of  the  header,  in  feet ;  m  and  k  the  length,  in 
feet,  respectively,  of  the  two  sets  of  tail  beams,  and  m  +  n  =  Tt 
+  g  =  l. 

The  constant  in  the  above,  (181,)  is  for  dwellings ;  if  the 
floor  is  to  be  loaded  more  than  dwelling  floors,  then  it  must  be 
increased  in  proportion  to  the  increase  of  load  up  to  as  high  as 
1600  for  warehouses. 

Rule  LIL — Trimmer  beams  for  framed  openings  occurring 
so  as  to  require  two  headers.  Multiply  the  square  of  the 
length  of  each  tail  beam  by  the  difference  of  length  of  the  tail 
beam  and  trimmer,  all  in  feet,  and  add  the  products ;  multiply 
their  sum  by  300  times  the  length,  in  feet,  of  the  header,  and 
divide  this  product  by  the  product  of  the  square  of  the  depth, 
in  inches,  by  the  length,  in  feet,  and  by  the  value  of  /#,  Table 
VI. ;  and  the  quotient,  added  to  the  thickness  of  a  common 
beam  of  the  tier,  will  equal  the  thickness  of  the  trimmer 
beams. 

Example. — A  tier  of  white  pine  beams,  4  X  14  inches,  20 
feet  long,  is  to  have  an  opening  of  5  x  10  feet,  framed  so  that 
the  length  of  one  series  of  tail  beams  is  7  feet,  the  other  8  feet, 
what  must  be  the  breadth  of  the  trimmers  ?  Here,  (7a  X  13) 
+  (8*  X  12)  equals  1405.  This  by  300  x  10  equals  4,215,000. 
This  divided  by  1,528,800  (=  14'  +  20  X  390)  equals  275,  and 
this  added  to  4,  the  breadth,  equals  6'75,  or  6|,  the  breadth 
required,  in  inches. 

345. — Additional  stiffness  is  given  to  a  floor  by  the  insertion 
of  bridging  strips,  or  struts,  as  at  a  a,  (Fig.  228.)  These  pre- 
vent the  turning  or  twisting  of  the  beams,  and  when  a  weight 
is  placed  upon  the  floor,  concentrated  over  one  beam,  they 
prevent  this  beam  from  descending  below  the  adjoining  beams 
to  the  injury  of  the  plastering  upon  the  underside.  It  is  usual 
to  insert  a  course  of  bridging  at  every  5  to  8  feet  of  the  length 


246  AMERICAN    HOUSE-CARPENTER. 


Fig.  228. 

of  the  beam.  Strips  of  board  or  plank  nailed  to  the  underside 
of  the  floor  beams  to  receive  the  lathing,  are  termed  cross- 
furring  ^  and  should  not  be  over  2  inches  wide,  and  placed  12 
inches  from  centres.  It  is  desirable  that  all  furring  be  narrow, 
in  order  that  the  clinch  of  the  mortar  be  interrupted  but  little. 
When  it  is  desirable  to  prevent  the  passage  of  sound,  the 
openings  between  the  beams,  at  about  3  inches  from  the  upper 
edge,  are  closed  by  short  pieces  of  boards,  which  rest  on  cleets, 
nailed  to  the  beam  along  its  whole  length.  This  forms  a  floor, 
on  which  mortar  is  laid  from  1  to  2  inches  deep.  This  is 
called  deafening. 

346. — When  the  distance  between  the  walls  of  a  building  is 
great,  it  becomes  requisite  to  introduce  girders,  as  an  addi- 
tional support,  beneath  the  beams.  The  dimensions  of  girders 
may  be  ascertained  by  the  general  rules  for  stiffness.  For- 
mulas (72,)  (73,)  and  (74,)  Table  Y.,  are  applicable,  taking  /, 
at  86,  for  dwellings  and  ordinary  stores,  and  increased  in  pro- 
portion to  the  load,  up  to  500,  for  heavy  warehouses.  When 
but  one  girder  occurs,  in  the  length  of  the  beam,  the  distance 
from  centres,  c,  is  equal  to  one-half  the  length  of  the  beam. 

347. — When  the  breadth  of  a  girder  is  more  than  about  12 
inches,  it  is  recommended  to  divide  it  by  sawing  from  end  to 
end,  vertically  through  the  middle,  and  then  to  bolt  it  to 


FRAMING.  247 

gether  with  the  sawn  sides  outwards.  This  is  not  to  strength- 
en the  girder,  as  some  have  supposed,  but  to  reduce  the  size 
of  the  timber,  in  order  that  it  may  dry  sooner.  The  opera- 
tion affords  also  an  opportunity  to  examine  the  heart  of  the 
stick — a  necessary  precaution;  as  large  trees  are  frequently 
in  a  state  of  decay  at  the  heart,  although  outwardly  they  are 
seemingly  sound.  When  the  halves  are  bolted  together,  thin 
slips  of  wood  should  be  inserted  between  them  at  the  several 
points  at  which  they  are  bolted,  in  order  to  leave  sufficient 
space  for  the  air  to  circulate  between.  This  tends  to  prevent 
decay ;  which  will  be  found  first  at  such  parts  as  are  not 
exactly  tight,  nor  yet  far  enough  apart  to  permit  the  escape 
of  moisture. 

348. — When  girders  are  required  .for  a  long  bearing,  it  is 
usual  to  truss  them  ;  that  is,  to  insert  between  the  halves  two 
pieces  of  oak  which  are  inclined  towards  each  other,  and 
which  meet  at  the  centre  of  the  length  of  the  gj'-dar,  like  the 
rafters  of  a  roof-truss,  though  nearly  if  not  quite  concealed 
within  the  girder.  This,  and  many  similar  methods,  though 
extensively  practised,  are  generally  worse  than  useless  ;  since 
it  has  been  ascertained  that,  in  nearly  all  such  cases,  the  ope- 
ration has  positively  weakened  the  girder. 

A  girder  may  be  strengthened  by  mechanical  contrivance, 
when  its  depth  is  required  to  be  greater  than  any  one  piece  oi 


2-18  AMERICAN    HOUSE-CARPENTER. 

timber  will  allow.  Fig.  229  shows  a  very  simple  yet  invalu 
able  method  of  doing  this.  The  two  pieces  of  which  the  gir- 
der is  composed  are  bolted,  or  pinned  together,  having  keys 
inserted  between  to  prevent  the  pieces  from  sliding.  The 
keys  should  be  of  hard  wood,  well  seasoned.  The  two  pieces 
should  be  about  equal  in  depth,  in  order  that  the  joint  be- 
tween them  may  be  in  the  neutral  line.  (See  Art.  317.)  The 
thickness  of  the  keys  should  be  about  half  their  breadth,  and 
the  amount  of  their  united  thicknesses  should  be  equal  to  a 
trifle  over  the  depth  and  one-third  of  the  depth  of  the  girder. 
Instead  of  bolts  or  pins,  iron  hoops  are  sometimes  used  ;  and 
when  they  can  be  procured,  they  are  far  preferable.  In  this 
case,  the  girder  is  diminished  at  the  ends,  and  the  hoops 
driven  from  each  end  towards  the  middle. 

349. — Beams  may  be  spliced,  if  none  of  a  sufficient  length 
can  be  obtained,  though  not  at  or  near  the  middle,  if  it  can 
be  avoided.  (See  Art.  281.)  Girders  should  rest  from  9  to 
12  inches  on  the  wall,  and  a  space  should  be  left  for  the  air 
to  circulate  around  the  ends,  that  the  dampness  may  evapo- 
rate. Floor-timbers  are  supported  at  their  ends  by  walls  of 
considerable  height.  They  should  not  be  permitted  to  rest 
upon  intervening  partitions,  which  are  not  likely  to  settle  as 
much  as  the  walls ;  otherwise  the  unequal  settlements  will 
derange  the  level  of  the  floor.  As  all  floors,  however  well- 
constructed,  settle  in  some  degree,  it  is  advisable  to  frame 
the  beams  a  little  higher  at  the  middle  of  the  room  than  at 
its  sides, — as  also  the  ceiling-joists  and  cross-furring,  when 
either  are  used.  In  single  floors,  for  the  same  reason,  the 
rounded  edge  of  the  stick,  if  it  have  one,  should  be  placed 
uppermost. 

If  the  floor-plank  are  laid  down  temporarily  at  first,  and  left 
to  season  a  few  months  before  they  are  finally  driven  together 
and  secured,  the  joints  will  remain  much  closer.  But  if  the 
edges  of  the  plank  are  planed  after  the  first  laying,  they  will 


249 

shrink  again ;  as  it  is  the  nature  of  wood  to  shrink  after  every 
planing  however  dry  it  may  have  been  before. 


PARTITIONS. 


350. — Too  little  attention  has  been  given  to  the  construction 
of  this  part  of  the  frame-work  of  a  house.  The  settling  of 
floors  and  the  cracking  of  ceilings  and  walls,  which  disfigure 
to  so  great  an  extent  the  apartments  of  even  our  most  costly 
houses,  may  be  attributed  almost  solely  to  this  negligence.  A 
square  of  partitioning  weighs  nearly  a  ton,  a  greater  weight, 
when  added  to  its  customary  load,  such  as  furniture,  storage, 
&c.,  than  any  ordinary  floor  is  calculated  to  sustain.  Hence 
the  timbers  bend,  the  ceilings  and  .cornices  crack,  and  the 
whole  interior  part  of  the  house  settles ;  showing  the  necessity 
for  providing  adequate  supports  independent  of  the  floor- 
timbers.  A  partition  should,  if  practicable,  be  supported  by 
the  walls  with  which  it  is  connected,  in  order,  if  the  walls  set- 
tle, that  it  may  settle  with  them.  This  would  prevent  the 
separation  of  the  plastering  at  the  angles  of  rooms.  For  the 
same  reason,  a  firm  connection  with  the  ceiling  is  an  im- 
portant object  in  the  construction  of  a  partition. 

351. — The  joists  in  a  partition  should  be  so  placed  as  to  dis- 
charge the  weight  upon  the  points  of  support.  All  oblique 
pieces  in  a  partition,  that  tend  not  to  this  object,  are  much 
better  omitted.  Fig.  230  represents  a  partition  having  a  door 
in  the  middle.  Its  construction  is  simple  but  effective.  Fig. 
231  shows  the  manner  of  constructing  a  partition  having  doors 
near  the  ends.  The  truss  is  formed  above  the  door-heads, 
and  the  lower  parts  are  suspended  from  it.  The  posts,  a  arid 
5,  are  halved,  and  nailed  to  the  tie,  c  d,  and  the  sill,  ef.  The 
braces  in  a  trussed  partition  should  be  placed  so  as  to  form,  aa 
near  as  possible,  an  angle  of  40  degrees  with  the  horizon.  In 
partitions  that  are  intended  to  support  only  their  own  weight, 
32 


250 


AMERICAN   HOUSE-CARPENTER. 


n 


u 


Fig.  230. 


Fig.  231. 


the  principal  timbers  may  be  3  X  4  inches  for  a  20  feet  span, 
3£  X  5  for  30  feet,  and  4x6  for  40.  The  thickness  of  the 
filling-in  stuff  may  be  regulated  according  to  what  is  said  at 
Art.  345,  in  regard  to  the  width  of  furring  for  plastering. 
The  filling-in  pieces  should  be  stiffened  at  about  every  three 
feet  by  short  struts  between. 

All  superfluous  timber,  besides  being  an  unnecessary  load 
upon  the  points  of  support,  tends  to  injure  the  stability  of  the 
plastering ;  for,  as  the  strength  of  the  plastering  depends,  in  a 
great  measure,  upon  its  clinch,  formed  by  pressing  the  mortar 


FRAMING.  251 

through  the  space  between  the  laths,  the  narrower  the  surface, 
therefore,  upon  which  the  laths  are  nailed,  the  less  will  be  the 
quantity  of  plastering  unclinched,  and  hence  its  greater  secu- 
rity from  fractures.  For  this  reason,  the  principal  timbers  of 
the  partition  should  have  their  edges  reduced,  by  chamfering 
off  the  corners. 


LJ 


352. — When  the  principal  timbers  of  a  partition  require  to 
be  large  for  the  purpose  of  greater  strength,  it  is  a  good  plan 
to  omit  the  upright  filling-in  pieces,  and  in  their  stead,  to 
place  a  few  horizontal  pieces ;  in  order,  upon  these  and  the 
principal  timbers,  to  nail  upright  battens  at  the  proper  dis- 
tances for  lathing,  as  in  Fig.  232.  A  partition  thus  con- 
structed requires  a  little  more  space  than  others ;  but  it  has 
the  advantage  of  insuring  greater  stability  to  the  plastering, 
and  also  of  preventing  to  a  good  degree  the  conversation  of 
one  room  from  being  heard  in  the  other.  When  a.  partition  is 
required  to  support,  in  addition  to  its  own  weight,  that  of  a 
floor  or  some  other  burden  resting  upon  it,  the  dimensions  of 
the  timbers  may  be  ascertained,  by  applying  the  principles 
which  regulate  the  laws  of  pressure  and  those  of  the  resistance 
of  timber,  as  explained  at  the  first  part  of  this  section.  The 
following  data,  however,  may  assist  in  calculating  the  amount 
of  pressure  upon  partitions  : 


252  AMERICAN   HOCSE-CARPENTER. 

Wh.te  pine  timber  weighs  from  22  to  32  pounds  per  cubic 
foot,  varying  in  accordance  with  the  amount  of  seasoning  it 
has  had.  Assuming  it  to  weigh  30  pounds,  the  weight  of  the 
beams  and  floor  plank  in  every  superficial  foot  of  the  flooring 
will  be,  when  the  beams  are 

8x8  inches,  and  placed  20  inches  from  centres,  6  pounds. 
3  x  10      "        "        "       18      "        "     .    "        Ti      " 
3  x  12      "         "         "        16      "         "  9 

3  x  12      "         "         "       12      "         "          «       11 

4  x  12      "         «         "       12      "         "          "       13 

4  x  14      "         "         "        14      "        "          "       13        " 

In  addition  to  the  beams  and  plank,  there  is  generally  the 
plastering  of  the  ceiling  of  the  apartments  beneath,  and  some- 
times the  deafening.  Plastering  may  be  assumed  to  weigh  9 
pounds  per  superficial  foot,  and  deafening  11  pounds. 

Hemlock  weighs  about  the  same  as  white  pine.  A  parti- 
tion of  3  x  4  joists  of  hemlock,  set  12  inches  from  centres, 
therefore,  will  weigh  about  2£  pounds  per  foot  superficial,  and 
when  plastered  on  both  sides,  20£  pounds. 

353. — When  floor  beams  are  supported  at  the  extremities, 
and  by  a  partition  or  girder  at  any  point  between  the  extre- 
mities, one-half  of  the  weight  of  the  whole  floor  will  then  be 
supported  by  the  partition  or  girder.  As  the  settling  of  parti- 
tions and  floors,  which  is  so  disastrous  to  plastering,  is  fre- 
quently owing  to  the  shrinking  of  the  timber  and  to  ill-made 
joints,  it  is  very  important  that  the  timber  be  seasoned  and 
the  work  well  executed.  Where  practicable,  the  joists  of  a 
partition  ought  to  extend  down  between  the  floor  beams  tc 
the  plate  of  the  partition  beneath,  to  avoid  the  settlement  con- 
sequent upon  the  shrinkage  of  the  floor  beams. 

BOOFS.* 
35i. — In  ancient  biildings,  the  Norman  and  the  Gothic,  the 

•  See  also  Art.  238. 


FRAIOHO. 


258 


walls  and  buttresses  were  erected  so  massive  and  firm,  that  il 
was  customary  to  construct  their  roofs  without  a  tie-heam : 
the  Avails  being  abundantly  capable  of  resisting  the  lateral 
pre^ure  exerted  by  the'  rafter?.  But  in  modern  buildings,  the 
walls  are  so  slightly  built  as  to  be  incapable  of  resisting 
scarcely  any  oblique  pressure;  and  hence  the  necessity  of 
constructing  the  roof  so  that  all  oblique  and  lateral  strains 
may  be  removed;  as,  also,  that  instead  of  having  a  tendency 
to  separate  the  walls,  the  roof  may  contribute  to  bind  and 
steady  them. 

355. — In  estimating  the  pressures  upon  any  certain  roof,  for 
the  purpose  of  ascertaining  the  proper  sizes  for  the  timbers, 
calculation  must  be  made  for  the  pressure  exerted  by  the  wind, 
and,  if  in  a  cold  climate,  for  the  weight  of  snow,  in  addition  to 
the  weight  of  the  materials  of  which  the  roof  is  composed. 
The  weight  of  snow  will  be  of  course  according  to  the  depth 
it  acquires.  Snow  weighs  8  Ibs.  per  cubic  foot,  and  more 
when  saturated  with  water.  In  a  severe  climate,  roofs  ought 
to  be  constructed  steeper  than  in  a  milder  one,  in  order  that 
the  snow  may  have  a  tendency  to  slide  off  before  it  becomes 
of  sufficient  weight  to  endanger  the  safety  of  the  roof.  The 
inclination  should  be  regulated  in  accordance  with  the  qualities 
of  the  material  with  which  the  roof  is  to  be  covered.  The 
following  table  may  be  useful  in  determining  the  smallest  in- 
clination, and  in  estimating  the  weight  of  the  various  kinds 
of  covering : 


Material 

Inclination. 

Weight  upon  a 
square  foot 

Tin 

Rise  1  inch  to  a  foot 

f  to  1}  Ibs. 

Copper          
Lenil     

1    to  H 
4    to  7 

Zinc                .         .         .         .         . 
Short  pine  shingles       .... 
Long  cypress  shingles  .... 
Slate    "  

"    8      "             " 
"    5      "             " 
"    6      " 

1      6      ' 

l{tol 

a  to  s 

5    to9 

254  AMERICAN   HO  TSE-CARPENTEB. 

The  weight  of  the  covering,  as  above  estimated,  is  that  of 
the  material  only,  with  the  weight  of  whatever  is  used  to  fix 
it  to  the  roof,  such  as  nails,  &c.  What  the  material  is  laid 
on,  such  as  plank,  boards  or  lath,  is  not  included.  The  weight 
of  plank  is  about  3  pounds  per  foot  superficial ;  of  boards,  2 
pounds;  and  lath,  about  a  half  pound. 

356. — The  weights  and  pressures  on  a  roof  arise  from  the 
roofing,  the  truss,  the  ceiling,  wind  and  snow,  and  may  be 
stated  as  follows : 

First,  the  Roofing. — On  each  foot  superficial  of  the  inclined 
surface, 

Slating will  weigh  about  7  Ibs. 

Roof  plank,  \\  inches  thick  .  "          "       2'T  " 

Roof  beams  or  jack  rafters     .  "         "          "       2'3  " 

Total,  12  Ibs. 

This  is  the  weight  per  foot  on  the  inclined  surface ;  but  it  is 
desirable  to  know  how  much  per  foot,  measured  horizontally, 
this  is  equal  to.  The  horizontal  measure  of  one  foot  of  the 
inclined  surface  is  equal  to  the  cosine  of  the  angle  of  inclina- 
tion. Therefore, 

cos.  :  1  ::  p  :  w  —  •+— ; 

COS. 

where  p  represents  the  pressure  on  a  foot  of  the  inclined  sur- 
face, and  w  the  weight  of  the  roof  per  foot,  measured  horizon- 
tally. The  cosine  of  an  angle  is  equal  to  the  base  of  the  right- 
angled  triangle  divided  by  the  hypothenuse,  which  in  this 
case  would  be  half  the  span  divided  by  the  length  of  the 

rafter,  or   — ,  where  s  is  the  span,  and  I  the  length  -f  the 

2  // 

rafter.    Hence, 

P         P   _  ^  IP 
^  ~  ~fi   ~    ~T"y 
or,  twice  the  pressure  per  foot  of  inclined  surface,  multiplied 


FRAMING.  255 

by  the  length  of  the  rafter,  and  divided  by  the  span,  will  give 
the  weight  per  foot  measured  horizontally;  or, 

24  -  =  w  (182.) 

s  , 

equals  the  weight  per  foot,  measured  horizontally,  of  the  roof 
beams,  plank,  and  covering  for  a  slate  roof. 

Second)  the  Truss. — The  weight  of  the  framed  truss  is  nearly 
in  proportion  to  the  length  of  the  truss,  and  to  the  distance 
apart  at  which  the  trusses  are  placed. 

w  =  5-2ca  (183.) 

equals  the  weight,  in  pounds,  of  a  white  pine  truss  with  iron 
suspension  rods  and  a  horizontal  tie  beam,  near  enough  foi 
the  requirements  of  our  present  purpose ;  where  s  equals  the 
length  or  span  of  the  truss,  and  c-the  distance  apart  at  which 
the  trusses  are  placed,  both  in  feet.  It  is  desirable  to  know 
how  much  this  is  equal  to  per  foot  of  the  area  over  which  the 
truss  is  to  sustain  a  covering.  This  is  found  by  dividing  the 
weight  of  the  truss  by  the  span,  and  by  the  distance  aoart 
from  centres  at  which  the  trusses  are  placed  ;  or, 

5^-*  =  5-2=W  (184) 

cs 

equals  the  weight  in  pounds  per  foot  to  be  allowed  for  the 
truss. 

Third,  the  Ceiling. — The  weight  supported  by  the  tie  beams, 
viz. :  that  of  the  ceiling  beams,  furring  and  plastering,  is  about 
9  pounds  per  superficial  foot. 

Fourth,  the  Wind. — The  force  of  wind  has  been  known  as 
high  as  50  pounds  per  superficial  foot  against  a  vertical  sur- 
face. The  effect  of  a  horizontal  force  on  an  inclined  surface 
is  in  proportion  to  the  sine  of  the  angle  of  inclination,  the  ef- 
fect produc'ed  being  in  the  direction  at  right  angles  to  the  in- 
clined surface.  The  force  thus  acting  may  be  resolved  into 
forces  acting  in  two  directions — the  one  horizontal,  the  other 
vertical ;  the  former  tending,  in  the  case  of  a  roof,  to  thrust 


256  AMERICAN   HOUSE-CARPENTER. 

aside  the  walls  on  which  the  roof  rests,  and  the  latter  acting 
directly  on  the  materials  of  which  the  roof  is  constructed— 
this  latter  force  being  in  proportion  to  the  sine  of  the  angle 
of  inclination  multiplied  by  the  cosine.  This  is  the  vertical 
effect  of  the  wind  upon  a  roof,  without  regard  to  the  surface 
it  acts  upon.  The  wind,  acting  horizontally  through  one  foot 
superficial  of  vertical  section,  acts  on  an  area  of  inclined  sur- 
face equal  to  the  reciprocal  of  the  sine  of  inclination,  and  the 
horizontal  measurement  of  this  inclined  surface  is  equal  to  the 
cosine  of  the  angle  of  inclination  divided  by  the  sine.  This  is 
the  horizontal  measurement  of  the  inclined  surface,  and  the 
vertical  force  acting  on  this  surface  is,  as  above  stated,  in  pro- 
portion to  the  sine  multiplied  by  the  cosine.  Combining  these, 
it  is  found  that  the  vertical  power  of  the  wind  is  in  proportion 
to  the  square  of  the  sine  of  the  angle  of  inclination.  There- 
fore, if  the  power  of  wind  against  a  vertical  surface  be  taken 
at  50  pounds  per  superficial  foot,  then  the  vertical  effect  on  a 
roof  is  equal  to 

w  =  50  sin.8  =  50  ^  (185.) 

for  each  piece  of  the  inclined  surface,  the  horizontal  measure- 
ment of  which  equals  one  foot ;  where  I  equals  the  length  of 
the  rafter,  and  h  the  height  of  the  roof. 

Fifth,  Snow.— The  weight  of  snow  will  be  in  proportion  to 
the  depth  it  acquires,  and  this  will  'be  in  proportion  to  the 
rigour  of  the  climate  of  the  place  at  which  the  building  is  to 
be  erected.  Upon  roofs  of  most  of  the  usual  inclinations, 
snow,  if  deposited  in  the  absence  of  wind,  will  not  slide  off. 
When  it  has  acquired  some  depth,  and  not  till  then,  it  will 
have  a  tendency,  in  proportion  to  the  angle  of  inclination,  to 
slide  off  in  a  body.  The  weight  of  snow  may  be  taken,  there- 
fore, at  its  weight  per  cubic  foot,  8  pounds,  multiplied  by  the 
depth  it  is  usual  for  it  to  acquire.  This,  in  the  latitude  of  New 
York,  may  be  stated  at  about  2£  feet.  Its  weight  would, 


FRAMING.  257 

therefore,  be  20  pounds  per  foot  superficial,  measured  horizon- 
tally. 

357. — There  is  one  other  cause  of  strain  upon  a  roof;  namely, 
the  load  that  may  be  deposited  in  the  roof  when  used  as  a  rooir 
for  storage,  or  for  dormitories.  But  this  seldom  occurs.  When 
a  case  of  this  kind  does  occur,  allowance  is  to  be  made  for  it 
as  shown  in  the  article  on  floors.  But  in  the  general  rule,  now 
under  consideration,  it  may  be  omitted. 

358. — The  following,  therefore,  comprehends  all  the  pressures 
or  weights  that  occur  on  roofs  generally,  per  foot  superficial ; 

For  roof  beams,  plank,  and  slate  (182)        .        .        .        .        34  jibs.      . 

"    the  truss  (184)        . 5'2    " 

"    ceiling 9       " 

"    wind  (185) .         .         .         60  ya " 

"    snow,  latitude  of  S" ew  York 20      " 

Having  found  the  weight  per  foot,  the  total  weight  for  any 
part  of  the  roof  is  found  by  multiplying  the  weight  per  foot  by 
the  area  of  that  part.  This  process  will  give  the  weight  sup- 
ported by  braces  and  suspension  rods,  and  also  that  supported 
by  the  rafters  and  tie  beam.  But  in  these  last  two,  only  half 
of  the  pressure  of  the  wind  is  to  be  taken,  for  the  wind  will 
act  only  on  one  side  of  the  roof  at  the  same  time. 

The  vertical  pressure  on  the  head  of  a  brace,  then,  equals 

W  =  4:  c  n(6  -  +  8-55  +  12-5  ^)  (186.) 

And     W  =  cp  n,  where  p  equals  4^6  -  -f  8*55  +  12-5  ^j, 

equals  the  weight  per  foot. 

And  the  aggregate  load  of  the  roof  on  each  truss  equals 

W  =  4  c  s  (G  -  +  8-55  +  6-25  ^)  (187.) 

N       6'  v   ' 

And     W  =  c q s,  where  q  =  4^6 -  +  S'55  +•  6-25  j^  equals  the 
33 


258  AMERICAN   HOUSE-CARPENTER. 

weight  per  foot ;  where  c  equals  the  distance  apart  from  centres 
at  which  the  trusses  are  placed ;  n  the  distance  horizontally 
between  the  heads  of  the  braces,  or,  if  these  are  not  located  at 
equal  distances,  then  n  is  the  distance  horizontally  from  a  point 
half-way  to  the  next  brace  on  one  side  to  a  point  half-way  to 
the  next  brace  on  the  other  side ;  I  the  length  of  the  rafter ;  s 
the  span,  and  Ji  the  height — all  in  feet. 

359. — By  the  parallelogram  of  forces,  the  weight  of  the  roof 
is  in  proportion  to  the  oblique  thrust  or  pressure  in  the  axis  of 
the  rafter,  as  twice  the  height  of  the  roof  is  to  the  length  of  the 
rafter;  or, 

W  :  R  ::  2  h  :  I,  or 

W7 

2A  :    I  ::W  :  £  =  fp  (188.) 

where  R  equals  the  pressure  in  the  axis  of  the  rafter.  And 
the  weight  of  the  roof  is  in  proportion  to  the  horizontal  thrust 
in  the  tie  beam,  as  twice  the  height  of  the  roof  is  to  half  the 
span;  or, 

W  :  H:\  2  A  : -1,  or 

2i  , 

2A  :-!::  W  :  H=^  (189.) 

where  II  equals  the  horizontal  thrust  in  the  tie  beam ;  the 
value  of  W  in  (188)  and  (189)  being  shown  at  (187),  and  (187) 
being  compounded  as  explained  in  Art,  356.  The  weight 
is  that  for  a  slate  roof.  If  other  material  is  used  for  covering, 
or  should  there  be  other  conditions  modifying  the  weight  in 
any  particular  case,  an  examination  of  Art.  356  will  show  how 
to  modify  the  formula  accordingly. 

360. — The  pressures  may  be.  obtained  geometrically,  as 
shown  in  Fig.  233,  where  A  B  represents  the  axis  of  the  tie 
beam,  A  C  the  axis  of  the  rafter,  D  E  and  FB  the  axes  of  the 
braces,  and  D  G,  FE,  and  C  B,  the  axes  of  the  suspension  rodg. 
In  this  design  for  a  truss,  the  distance  A  B  is  divided  into  three 


FRAMING.  259 

equal  parts,  and  the  rods  located  at  the  two  points  of  division, 
G  and  E.  By  this  arrangement  the  rafter  Ads  supported  at 
equi-distant  points,  D  and  F.  The  point  D  supports  the  rafter 
for  a  distance  extending  half-way  to  A  and  half-way  to  F^  and 
the  point  F  sustains  half-way  to  D  and  half-way  to  C.  Also, 
the  point  C  sustains  half-way  to  F  and,  on  the  other  rafter, 
half-way  to  the  corresponding  point  to  F.  And  because  these 
points  of  support  are  located  at  equal  distances  apart,  there- 
fore the  load  on  each  is  the  same  in  amount.  On  D  G  make 
D  a  equal  to  100  of  any  decimally  divided  scale,  and  let  D  a 
represent  the  load  on  Z>,  and  draw  the  parallelogram  a~bDc. 
Then,  by  the  same  scale,  D  b  represents  (Art.  258)  the  pressure 
in  the  axis  of  the  rafter  by  the  load  at  D  ;  also,  D  c  the  pressure 
in  the  brace  D  E.  Draw  c  d  horizontal ;  then  D  d  is  the  ver- 
tical pressure  exerted  by  the  brace  DE  at  -El  The  point  F 
sustains,  besides  the  common  load  represented  by  100  of  the 
scale,  also  the  vertical  pressure  exerted  by  the  brace  D  E; 
therefore,  since  D  a  represents  the  common  load  on  D,  F^  or 
C,  make  Fe  equal  to  the  sum  of  Da  and  D  d,  and  draw  the 
parallelogram  Fgef.  Then  Fg,  measured  by  the  scale,  is 
the  pressure  in  the  axis  of  the  rafter  caused  by  the  load  at  F, 
and  Ff  is  the  load  in  the  axis  of  the  brace  F B.  Draw  fh 
horizontal ;  then  Fh  is  the  vertical  pressure  exerted  by  the 
brace  F  B  at  B.  The  point  £7,  besides  the  common  load  re- 
presented by  D  a,  sustains  the  vertical  pressure  Fh  caused  by 
the  brace  F  B.  and  a  like  amount  from  the  corresponding:  brace 
on  the  opposite  side.  Therefore,  make  Cj  equal  to  the  sum  of 
Da  and  twice  Fh,  and  draw  jk  parallel  to  the  opposite  rafter. 
Then  Ck  is  the  pressure  in  the  axis  of  the  rafter  at  G.  This  is 
not  the  only  pressure  in  the  rafter,  although  it  is  the  total 
pressure  at  its  head  C.  At  the  point  F,  besides  the  pressure 
C k,  there  is  Fg.  At  the  point  D,  besides  these  two  pressures, 
there  is  the  pressure  D  I.  At  the  foot,  at  J.,  there  is  still  an 
additional  pressure  :  while  the  point  D  sustains  the  load  half- 


260  AMERICAN   HOUSE-CARPENTER. 

way  to  F  and  half-way  to  A^  the  point  A  sustains  the  load 
half-way  to  D.  This  load  is,  in  this  case,  just  half  the  load  at 
D.  Therefore  draw  A  m  vertical,  and  equal  to  50  of  the  scale, 
or  half  of  D  a.  Extend  C  A  to  I  /  draw  m  I  horizontal.  Then 
A  I  is  the  pressure  in  the  rafter  at  A  caused  by  the  weight  of 
tlie  roof  from  A  half-way  to  D.  Now  the  total  of  the  pressures 
in  the  rafter  is  equal  to  the  sum  of  A  I  +  D  b  -f-  Fg  added  to 
Ck.  Therefore  make  ~k  n  equal  to  the  sum  of  A I  4-  D I  +  Fg, 
and  draw  n  o  parallel  with  the  opposite  rafter,  and  nj  hori- 
zontal. Then  Co,  measured  by  the  same  scale,  will  be  found 
equal  to  the  total  weight  of  the  roof  on  both  sides  of  C  B.  If 
Da  =  100  represent  the  portion  of  the  weight  borne  by  the 
point  Z>,  then  Co,  representing  the  whole  weight  of  the  roof, 
should  equal  600,  (as  it  does  by  the  scale,)  for  D  supports  just 
one-sixth  of  the  whole  load.  As  Cn  is  the  total  oblique  thrust 
in  the  axis  of  the  rafter  at  its  foot,  therefore  nj  is  the  horizon- 
tal thrust  in  the  tie  beam. 

361. — In  stating  the  amount  of  pressures  in  the  above  as 
being  equal  to  certain  lines,  it  was  so  stated  with  the  under- 
standing that  the  lines  were  simply  in  proportion  to  the  weights. 
To  obtain  the  weight  represented  by  a  line,  multiply  its  length 
(measured  by  the  scale  used)  by  the  load  resting  at  D,  (or  at 
F  or  (7,  as  these  are  all  equal  in  this  example,)  and  divide  the 
product  by  100,  and  the  quotient  will  be  the  weight  required. 
For,  as  100  of  the  scale  is  to  the  load  it  represents,  so  is  any 
other  dimension  on  the  same  scale  to  the  load  it  represents. 

362.— -Example.  Let  A  B  (Fig.  233)  equal  26  feet,  CB  13 
feet,  and  A  C  29  feet,  and  A  G,  G  E,  and  EB,  each  8|  feet. 
Let  the  trusses  be  placed  10  feet  apart.  Then  the  weight  on 
D,  for  the  use  of  the  braces  and  rods,  is,  per  (186),  equal  to 

4c»(6^'f  8-55  +  121^) 

=  4:  X  10  x  82(6  x  |?  +  8-55  +  12*  X  |i| 
\       y2i  zy 


FRAMING.  20;. 

=  346|  X  14-398 
=  4991-3. 

This  is  the  common  load  at  the  points  D,  F,  and  C,  and  each 
of  the  lines  denoting  pressures  multiplied  by  it  and  divided  by 

4991  "3 

100,  or  multiplied  by  the  quotient  of  — =.  49-913.  the  pro- 
duct will  be  the  weight  required.  49*913  may  be  called  50, 
for  simplicity  ;  therefore  the  pressure  in  the  brace  D  E  equals 
112  X  50  =  5600  pounds,  and  in  the  brace  F  B,  140  X  50  - 
7000  pounds,  and  in  like  manner  for  any  other  strain.  For 
the  rafters  and  tie  beam  the  total  weight,  as  per  (187),  equals 

4cs(6-  +  8-55  +  6i4) 

=  4  x  10  x  52(6  x  —  +  8-55  +  6J  X  |S) 

=  2080  x  13-148 
=  27343-68  pounds. 

This  is  the  total  weight  of  the  roof  supported  by  one  truss. 
The  oblique  thrust  in  the  rafter  A  0  is,  per  (188),  equal  to 

l—=  29  x  27343-68 
2  h  2"x~HT~ 

=  30498-72  pounds. 

To  obtain  this  oblique  thrust  geometrically  :  Co  (Fig.  233) 
represents  the  weight  of  the  roof,  and  measures  600  by  the  scale  ; 
and  the  line  Cn,  representing  the  oblique  thrust,  measures  670. 
By  the  proportion,  600  :  670  ::  27343-68  :  30533-8,  =  the 
oblique  thrust.  The  result  here  found  is  a  few  pounds  more 
than  the  other.  This  is  owing  to  the  fact  that  the  line  Cn  is 
not  exactly  670,  nor  is  the  length  of  the  rafter  precisely  29  feet. 
Were  the  exact  dimensions  used  in  each  case  the  results  would 
be  identical ;  but  the  result  in  either  case  is  near  enough  foi 
the  purpose. 

The  horizontal  strain  is,  per  (189),  equal  to 


262  AMERICAN    IIOUSE-CAKPENTEE. 

Ws  _  27343-68  X  52 
~Th  ~~         4  X  13 

=  27343-68  pounds. 

The  result  gives  the  horizontal  thrust  precisely  equal  to  the 
weight.  This  is  as  it  should  be  in  all  cases  where  the  height 
of  the  roof  is  equal  to  one-fourth  of  the  span,  but  not  other- 
wise ;  for  the  result  depends  (189)  upon  this  relation  of  the 
height  to  the  span.  Geometrically)  the  result  is  the  same,  for 
Co  and  nj  (Fig.  233,)  representing  the  weight  and  horizontal 
thrust,  are  precisely  equal  by  measurement. 

363.  —  The  weight  at  the  head  of  a  brace  is  sustained  partly 
at  the  foot  of  the  brace  and  partly  at  the  foot  of  the  rafter. 
The  sum  of  the  vertical  effects  at  these  two  points  is  just  equal 
to  the  weight  at  the  head  of  the  brace.  The  portion  of  the 
weight  sustained  at  either  point  is  in  proportion,  inversely,  to 
the  horizontal  distance  of  that  point  from  the  weight  ;  there- 
fore, 

F=TF|,  (190.) 

where  V  equals  the  vertical  effect  at  the  foot  of  the  brace  ;  TP, 
the  weight  at  the  head  of  the  brace  ;  g,  the  horizontal  distance 
from  the  foot  of  the  rafter  to  the  head  of  the  brace  ;  and  «,  the 
distance  from  the  same  point  to  ft&foot  of  the  brace. 

364:.  —  For  the  oblique  thrust  in  the  brace  :  from  the  triangle 


Fh  :  Ff::  sin.  :  rad. 

sin.  :  rad.  ::  V  I  T\ 
therefore, 


(191.) 
h 


sin. 

where  T  equals  the  oblique  thrust  in  the  brace;  V,  the  verti- 
cal pressure  caused  by  T  at  the  foot  of  the  brace  (190)  ;  a 
I  and  h  the  length  and  height  respectively  of  the  brace. 

365.—  Example.     Brace-  D  E,  Fig.  233.     In  this  case, 
equals  the  product  of  the  weight  per  superficial  foot,  m 


FRAMING.  263 

plied  by  the  area  supported  at  the  point  D,  equals  5000 
pounds,  (Art.  362.)  The  length  g  equals  Sf  feet,  and  a  equals 
17i  feet.  Therefore  (190), 

V=  W^=  5000  x  -?!  =  2500  pomids 
a  17£ 

equals  vhe  vertical  pressure  at  E  caused  by  the  brace  D  E. 
Ther  for  the  oblique  thrust,  I  equals  9*6  feet,  and  h  equals  4'3 
feet.  Therefore,  from  (191), 

T  =  V\  =  2500  x  ^|  =  5581-4  pounds 
h  4*3 

equals  the  oblique  thrust  in  the  brace  D  E.  In  Art.  362  it 
was  found  to  be  5600.  The  discrepancy  is  owing  to  like  causes 
of  want  of  accuracy  in  the  case  of  the  rafter,  as  explained  .in 
Art.  362. 

Another  Example.— Brace  FB,  Fig.  233.  In  this  case,  W 
equals  the  product  of  the  weight  per  superficial  foot,  multiplied 
by  the  area  supported  by  the  point  F,  added  to  the  vertical 
strain  caused  by  the  brace  D  E.  From  Art.  362  the  weight 
of  roof  on  F  equals  5000  pounds,  and  the  vertical  strain  from 
brace  D  E  is,  as  just  ascertained,  =  2500,  total  7500,  equals 
W.  The  length,  g,  equals  two-thirds  of  26  feet,  equals  17^, 
and  a  equals  26  feet.  Therefore,  from  (190), 

n  174 

F=TF-£  =  7500  x  —  =  COOO 

equals  the  vertical  effect  at  B  caused  by  the  brace  FB. 
Then,  for  the  oblique  thrust  in  the  brace,  /  equals  12*2,  and  h 
equals  8f.  Therefore,  from  (191) 

7  19*9 

T=  F^  =  5000  x  ~  =  7038-5 

equals  the  oblique  thrust  or  strain  in  the  axis  of  the  brace.  It 
was  7000  by  the  geometrical  process,  (Art.  362.) 

366. — The  strain  upon  the  first  rod,  D  G,  equals  simply  the 
weight  of  the  ceiling  supported  by  it,  added  to  the  part  of  the 
tie  beam  it  sustains.  The  weight  of  the  tie  beam  will  equal 


264 


AMEEICAN   HOUSE-CAKPENTER. 


FKAilLN'G.  265 

about  one  pound  per  superficial  foot  of  the  ceiling.  The  weighi 
per  foot  for  the  ceiling  is  stated  (see  Art.  356  third,  and  358,) 
at  9  pounds.  To  this  add  1  pound  for  the  tie  beam,  and  the 
sum  is  10.  Then 

jy=10cn.  (192.) 

The  strain  on  the  second  tie  rod  equals  the  .weight  of  ceiling 
supported,  —  N,  added  to  the  vertical  effect  of  the  strain  in  the 
brace  it  sustains,  [see  (190)]  or,  equal  to 

0  =  Wcn  +  V.  (193.) 

The  strain  on  the  third  rod  is  equal  to  N~,  added  to  the  ver- 
tical effect  of  the  strain  in  the  brace  it  sustains,  and  this  is  the 
strain  on  any  rod.  The  first  rod  has  no  brace  to  sustain,  and 
the  middle  rod  sustains  two  braces.  In  this  case  the  strain 
equals 

Z7=10c7i-+ 2F.  (191.) 

It  may  be  observed  that  V  represents  the  vertical  strain 
caused  by  that  brace  that  is  sustained  by  the  rod  under  consi- 
deration ;  and,  as  the  vertical  strain  caused  by  any  one  brace  is 
more  than  that  caused  by  any  other  brace  nearer  the  foot  of 
the  rafter,  therefore  the  V  of  (193)  is  not  equal  to  the  V  of 
(194):  Hence  a  necessity  for  care  lest  the  two  be  confounded 
and  thus  cause  error. 

367.— Examples.  The  rod  D  G  (Fig.  233)  has  a  strain 
which  equals  (192) 

N  =  .10  c  n  =  10  x  10  x  8|  =  867  pounds. 

The  strain  on  rod  FE  equals  (193) 

O  =  10  c  n  +  V  =  867  +  2500  -  3367  pounds. 

The  strain  on  rod  C  B,  the  middle  rod,  equals  (194) 
U  =  10  c  n  +  2  V=  867  +  2  x  5000  =  10867  pounds. 

368. — The  load,  and  the  strains  caused  thereby,  having 
been  discussed,  it  remains  to  speak  of  the  resistance  of  the  ma- 
terials. 

first,  of  the  Rafter. — Generally  this  piece  of  timber  is  so 
pinioned  by  the  roof  beams  or  purlins  as  to  prevent  any  late- 


266  AMERICAN   I1OUSE-CARPENTEE. 

ral  movement,  and  the  braces  keep  it  from  deflection  ;  there 
fore  it  is  not  liable  to  yield  by  flexure.  Hence  the  manner  of 
its  yielding,  when  overloaded,  will  be  by  crushing  at  the  ends, 
or  it  will  crush  the  tie  beam  against  which  it  presses.  The 
fibres  of  timber  yield  much  more  readily  when  pressed  toge- 
ther by  a  force  acting  at  right  angles  to  the  direction  of  their 
length,  than  when  it  acts  in  a  line  with  their  length. 

The  value  of  timber  subjected  to  pressure  in  these  two  ways 
is  shown  in  Art.  292,  Table  I.,  the  value  per  square  inch  of 
the  first  stated  resistance  being  expressed  by  P,  and  that  of 
the  other  by  C.  Timber  pressed  in  an  oblique  direction  yields 
with  a  force  exceeding  that  expressed  by  P,  and  less  than  that 
by  C.  When  the  angle  of  inclination  at  which  the  force  acts 
is  just  45°,  then  the  force  will  be  an  average  between  P  and 
G.  And  for  any  angle  of  inclination,  the  force  will  vary  in- 
versely as  the  angle ;  approaching  P  as  the  angle  is  enlarged, 
and  approaching  C  as  the  angle  is  diminished.  It  will  be 
equal  to  C  when  the  angle  becomes  zero,  and  equal  P  when 
the  angle  becomes  90°.  The  resistance  of  timber  per  square 
inch  to  an  oblique  force  is  therefore  expressed  by 

M  =  P  +  —  (C  -  P\  (195.) 

where  A°  equals  the  complement  of  the  angle  of  inclination. 
In  a  roof,  A°  is  the  acute  angle  formed  by  the  rafter  with 
a  vertical  line.  If  no  convenient  instrument  be  at  hand  to 
measure  the  angle,  describe  an  arc  upon  the  plan  of  the 
truss — thus:  with  CB  (Fig.  233)  for  radius,  describe  the 
arc  B  g,  and  get  the  length  of  this  arc  by  stepping  it  off  with 
a  pair  of  dividers.  Then 

A°  a 

A.£O2 

90  A' 

where  a  equals  the  length  of  the  arc,  and  h  equals  B  C,  the 
height  of  the  roof.  Therefore, 


FRAMING.  267 

M  =  P  +  0-63f  |(<7  -  P}  (196.) 

equals  the  value  of  timber  per  square  inch  in  a  tie  beam,  C 
and  P  being  obtained  from  Tabfe  I,  Art.  292.  When  0  for 
the  kind  of  wood  in  the  tie  beam  exceeds  C  set  opposite  the 
kind  of  wood  in  the  rafter,  then  the  latter  is  to  be  used  in  the 
rules  instead  of  the  former. 

369.  —  Having  obtained  the  strain  to  which  the  material  is 
subjected  in  a  roof,  and  the  capability  of  the  material  to  resist 
that  strain,  it  only  remains  now  to  state  the  rules  for  determin- 
ing the  dimensions  of  the  material. 

370.  —  To  obtain  the  dimensions  of  the  rafter  :  —  It  has  been 
shown  that  the  strain  in  the  axis  of  the  rafter  equals  (1.88), 

R  =  W* 

2A 

This  is  the  strain  in  pounds.  Timber  is  capable  of  resisting 
effectually,  in  every  square  inch  of  the  surface  pressed  (196), 

P  +  0-63|  |  (O-P)  pounds. 
And  when  the  strain  and  resistance  are  equal, 


where  5  and  d  are  respectively  the  breadth  and  depth  of  the 
rafter.     Hence 

ld  =  --  -  -  .  (19T.) 

P  +  0-63f  I  (C  -P) 

Example.  —  (Fig.  233.)  The  strain  in  the  axis  of  the  rafter 
m  this  example,  ascertained  in  Art.  362,  is  30498*72  pounds. 
If  the  timber  used  be  white  pine,  then  P  =  300  and  C=  1200, 
The  length  of  the  arc  Bg  is  14£  feet,  and  h  =  13.  Therefore 

ld=  _  *M»™    __  =32-8. 
300  +  (0-63|  X  y  X  900) 

This  is  the  area  of  the  abutting  surface  at  the  tie  beam- 
say  6  by  5£  inches.  At  least  half  this  amount  should  be  added 


268  AMERICAN   HOUSE-CARPENTER. 

to  allow  for  the  shoulder,  and  for  cutting  at  the  joints  fo. 
braces,  &c.  The  rafter  may  therefore  be  6  by  9  inches. 

The  above  method  is  based  upon  the  supposition  that  the 
rafter  is  effectually  secured  from  flexure  by  the  braces  and 
roof  beams.  Should  this  not  be  the  case,  then  the  dimensions 
of  the  rafter  are  to  be  obtained  by  rules  in  Art.  298,  for  posts. 
Nevertheless,  the  abutting  surface  in  the  joint  is  to  be  deter- 
mined by  the  above  formula  (197). 

,  371. — To  obtain  the  dimensions  of  the  braces: — Usually, 
braces  are  so  slender  as  to  require  their  dimensions  to  be  ob- 
tained by  rules  in  Art.  298 ;  the  strain  in  the  axis  of  the  brace 
having  been  obtained  by  formula  (191),  or  geometrically  as  in 
Art.  360. 

The  abutting  surface  of  the  joint  of  the  brace  is  to  be  ob- 
tained, as  in  the  case  of  the  rafter,  b}7  formula  (195) ;  A°  be- 
ing the  number  of  degrees  contained  in  the  acute  angle  formed 
by  the  brace  and  a  vertical  line,  for  the  joint  at  the  tie  beam  ; 
but  for  the  joint  at  the  rafter,  A°  is  the  number  of  degrees 
contained  in  the  acute  angle  formed  by  the  brace  and  a  line 
perpendicular  to  the  rafter,  or  it  is  90,  diminished  by  the  num- 
ber of  degrees  contained  in  the  acute  angle  formed  by  the 
rafter  and  brace. 

Example. — Fig.  233,  Brace  D  E,  of  white  pine.  In  this 
brace  the  strain  was  found  (Art.  362)  to  be  5600  pounds,  the 
length  of  the  brace  is  9'6  feet.  By  Art.  298,  the  brace  is 
therefore  required  to  be  4'18  X  6  inches.  For  the  abutting 
surface  at  the  joints,  for  white  pine,  P  equals  300  and, -C 1200. 
The  angle  D  EF equals  63°  26'.  By  (197)  and  (195), 

1)d_      T      _ 5600 

~  P  +  %(C-P)  ~  300  +  [-°026' x  (1200 -  300)] 
5600 


. 
= 

This  is  the  area  of  the  abuttirg  surface  of  the  joint  at.  the  tio 


FRAMING.  269 

beam.     To  obtain   the  joint  at  the  rafter,  the  angle  F  D  F. 
equals  53°  8',  and  hence 

h  d T_ .__ 5600 

~  P  +  %(€—  P)  ~  300  +[?2^9r-§1  X  (1200  -  300)] 

—  8*375  inches. 


300  +       r  X  900) 

This  is  the  area  of  the  abutting  surface  of  the  joint  at  the 
rafter. 

Another  Example. — Brace  F 'JB,  Fig.  233,  of  white  pine, 
12-2  feet  long.  The  strain  in  its  axis  is  (Art.  362)  7000 
pounds.  By  Art.  298,  the  brace  is  required  to  be  5J  x  6 
inches.  For  the  abutting  surface  of  the  joints,  P  equals  300, 
C  equals  1200,  and  the  angle  2*  B  G  equals  45°  ;  therefore, 

7000 

o  a  = = =  9$  inches. 

300+ [|  x(1200-  300)] 

This  is  the  area  of  the  abutting  surface  at  the  tie  beam.  For 
the  surface  at  the  rafter,  the  angle  C  F  B  equals  71°,  and 
90  —  71  =  19,  equals  the  angle  to  be  used  in  the  formula ; 

therefore, 

'7000 

5  d  = ^ =  14'3  inches,  nearly. 

300  -H  [I  X  (1200-300)] 

This  is  the  area  of  the  abutting  surface  of  the  joint  at  the 
rafter. 

372. — To  obtain  the  dimensions  of  the  tie  beam : — A  tie 
beam  .must  be  of  such  dimensions  as  will  enable  it  to  resist 
effectually  the  tensile  strain  caused  by  the  horizontal  thrust  of 
the  rafter  and  the  cross  strains  arising  from  the  weight  of  the 
ceiling,  and  from  any  load  that  may  be  placed  upon  it  in  the 
roof.  From  (17),  Art.  310, 

w      H 

-T~T 

where  H  equals  the  horizontal  thrust,  and  from  (189), 


270  AMERICAN   HOUSE-CABPENTEB. 

77       Ws- 

=  T7T' 

therefore, 

H  _Ws 

-- 


where  TF  equals  the  weight  of  the  roof  in  pounds,  as  shown 
at  (187);  s,  the  span  ;  A,  the  height,  both  in  feet;  and  T,  a 
constant  set  opposite  the  kind  of  material,  in  Table  III.  ;  and 
A  equals  the  area  of  uncut  fibres  in  the  tie  beam.  About 
one-half  of  this  should  be  added  to  allow  for  the  requisite  cut- 
ting at  the  joints  ;  or,  the  area  of  the  cross  section  of  the  tie 
beam  should  be  equal  to  at  least  f  of  the  area  of  uncut  fibres  ; 
or,  when  5  d  equals  the  area  of  the  tie  beam,  then 


Example.  —  The  weight  on  the  truss  at  Fig.  233  is  shown  to 
be  (Art.  362)  27343-68  pounds,  say  27500  pounds  ;  the  span  is 
52  feet,  the  height  13,  and  the  value  of  T  for  white  pine  is 
(Table  III.)  2367,  therefore 


equals  the  area  of  cross  section  of  the  tie  beam  requisite  to 
resist  the  tensile  strain.  This  is  smaller,  as  will  be  shown,  than 
what  is  required  to  resist  the  cross  strains,  and  this  will  be 
found  to  be  the  case  generally.  The  weight  of  the  ceiling  is 
9  pounds  per  superficial  foot  ;  the  length  of  the  longest  unsup- 
ported part  of  the  tie  beam  is  8|  feet  ;  then,  if  the  deflection 
per  lineal  foot  be  allowed  at  O'Olo  inch,  the  depth  of  the  tie 
beam  will  be  required  ((72),  Table  Y.)  to  be  6'14  inches.  But 
in  order  effectually  to  resist  the  strains  tie  beams  are  subjected 
to  at  the  hands  of  the  workmen,  in  the  process  of  framing  and 
elevating,  the  area  of  cross  section  in  inches  should  be  at 
least  equal  to  the  length  in  feet.  Were  it  possible  to  guard 
against  this  cause  of  strain,  the  size  ascertained  by  the  rule,  6 


FRAMING.  1^1 

by  6-14,  would  be  sufficient  ;  but  to  resist  this  strain,  the  size 
should  be  6  by  9. 

There  is  yet  one  other  dimension  for  the  tie  beam  required, 
and  that  is,  the  distance  at  which  the  joint  for  the  rafter  must 
be  located  from  the  end  of  the  tie  beam,  in  order  that  the 
thrust  of  the  rafter  may  not  split  off  the  part  against  which  it 
presses.  This  may  be  ascertained  by  Rule  XL,  Art.  302,  for 
all  cases  where  no  iron  strap  or  bolt  is  used  to  secure  the  joint  ; 
but  where  these  fastenings  are  used  the  abutment  may  be  of 
any  convenient  length.  And  in  using  irons  here,  care  should 
be  exercised  to  have  the  surface  of  pressure  against  the  iron 
of  sufficient  area  to  prevent  indentation. 

373.  —  To  obtain  the  dimensions  of  the  iron  suspension  rods. 

By  Art.  310,  (IT), 

w 

A=r> 

and  T  varies  (Table  III.)  from  5000  to  17000,  according  to  the 
diameter  inversely  ;  for  the  smaller  rods  are  stronger  in  pro- 
portion than  the  larger  ones. 

Example.  —  Taking  T  equal  5000,  then  the  area  of  the  rod 
D  G  (Fig.  233)  requires  (Art.  367)  to  be  equal  to 


corresponding  to  0'469  inch  diameter.     This  rod  may  be  half 
inch  diameter. 

Another  Example.—  The  rod  FE  (Fig.  233)  is  loaded  with 
(Art.  367)  3367  pounds,  therefore 


equals  the  area  of  the  rod,  the  corresponding  diameter  of  which 
is  0'925.     This  rod  may  be  one  inch  diameter. 

Again,  a  third  example;  the  rod  C  B.    This  rod  is  loaded 
with  (Art.  367)  10867  pounds,  therefore 


AMERICAN   HOUSE-CARPENTER. 

10867 


equals  the  required  area  of  the  rod,  the  diameter  correspond- 
ing tD  which  is  1*66.  This  rod  may  therefore  be  If  inches 
diameter. 

374.  —  While  discussing  the  principles  of  strains  in  roofs  and 
deducing  rules  therefrom,  the  truss  indicated  in  Fig.  233  has 
been  examined  throughout.     The  result  is  as  follows  :  rafter, 
6x9;  tie  beam,  (6  X  6,  or)  6x9;  the  first  brace  from  the 
wall,  4£  x  6  inches,  with  an  abutting  surface  at  the  lower  end 
of  6  inches,  and  at  the  upper  end  of  8f  inches  ;  the  other 
brace,  5J  x  6  inches,  with  an  abutting  surface  at  the  lower 
end  of  9£  inches,  and  at  the  upper  end  of  14TV  inches  ;  the 
shortest  rod,  •£  inch  diameter  ;  the  next,  1  .inch  diameter  ;  and 
the  middle  rod,  If  inches  diameter. 

PRACTICAL   RULES    AND   EXAMPLES. 

For  Roofs  Loaded  as  per  Art.  356. 

375.  —  Rule  LIII.     To  obtain  the  dimensions  of  the  rafter 
Multiply  the  value  of  R  (Table  IX.,  Art.  376)  by  the  span 
of  the  roof,  by  the  length  of  the  rafter,  and  by  the  distance 
apart  from  centres  at  which  the  roof  trusses  are  placed,  all  in 
feet,  and  divide  the  product  by  the  sum  of  twice  the  height 
of  the  roof  multiplied  by  the  value  of  P,  Table  I.,  set  opposite 
the  kind  of  wood  used  in  the  tie  beam,  added  to  the  difference 
of  the  values  of  (7Aand  P  in  said  table  multiplied  by  \\  times 
the  length  of  the  arc  that  measures  the  acute  angle  formed 
between  the  rafter  and  a  vertical  line,  the  arc  having  the  height 
of  the  roof  for  radius  (see  arc  B  #,  Fig.  233),  and  the  quo 
tient  will  be  the  area  of  the  abutting  surface  of  the  joint  at 
the  foot  of  the  rafter.     To  the  abutting  surface  add  its  half, 
and  the  sum  will  be  the  area  of  the  cross  section  of  the  rafter 


FRAMING.  273 

This  rule  is  upon  the  presumption  that  the  rafter  is  secured 
from  flexure  by  the  roof  beams  and  by  braces  and  ties  at 
short  intervals,  as  in  Fig.  233.  In  roofs  where  the  rafter  does 
not  extend  up  to  the  ridge  of  the  roof  but  abuts  against  a 
horizontal  straining  beam  (0,  Fig.  237),  in  the  rule  for  rafters, 
take  for  the  length  of  the  rafter  the  distance  from  the  foot  of 
the  rafter  to  the  ridge  of  the  roof;  or,  a  distance  equal  to 
what  the  rafter  would  be  in  the  absence  of  a  straining  beam. 
The  area  of  cross  section  of  the  straining  beam  should  be  made 
equal  to  that  of  the  rafter,  as  found  by  the  rule  so  modified. 

Example. — Find  the  dimensions  of  a  rafter  for  a  roof  truss 
whose  span  is  52  feet,  and  height  13  ;  the  length  of  the  rafter 
being  29  feet,  the  trusses  placed  10  feet  apart  from  centres, 
and  the  arc  -measuring  the  angle  at  the  head  of  the  rafter 
(having  the  Tieight  of  the  roof  for  radius)  being  144  feet, 
white  pine  being  used  in  the  tie  beam.  The  height  of  this 
roof  being  in  proportion  to  the  span  as  1  to  4,  the  value  of  It 
in  Table  IX.  is  52*6 ;  multiplying  this,  in  accordance  with  the 
rule,  by  52,  the  span  of  the  roof,  and  by  29,  the  length  of  the 
rafter,  and  by  10,  the  distance  between  the  roof  trusses,  the 
product  is  793208.  The  value  of  P  for  white  pine  in  Table  I. 
is  300 ;  multiplying  this  by  2  x  13  =  26,  twice  the  height  of 
the  roof,  the  product  is  7800.  The  value  of  C  for  white  pine, 
(Table  I.)  is  1200,  hence  the  difference  of  the  values  of  C  and 
P  is  1200-300  =  900;  this  multiplied  by  H,  and  by  144, 
the  length  of  the  arc,  the  product  is  16031 ;  this  added  to  the 
7800  aforesaid,  the  sum  is  23831.  The  aforesaid  product  of 
793208,  divided  by  this  23831,  the  quotient,. 33'3,  equals  the 
area  in  inches  of  the  abutting  surface  of  the  joint  at  the  tie 
beam.  To  this  add  16-7,  its  half,  and  the  sum,  50,  equals  the 
area  of  cross  section  of  the  rafter.  This  divided  by  the  thick- 
ness of  the  rafter,  say  6  inches,  the  quotient,  84,  is  the  breadth. 
The  rafter  is  therefore  to  be  6  X  84  inches.  It  may  be  made 
6x9,  avoiding  the  fractions. 
>  35 


274 


AMERICAN   HOJSE-CARFENTER. 


376. — The  following  table,  calculated  upon  data  in  Art.  358. 
presents  the  weight  per  foot  for  roofs  of  various  inclinations, 
and  covered  with  slate. 


When  height  of  roof 

The  vertical  strain  per  foot  of  surface  supported,  measured  horizontally, 

Is  to  span  as 

on  rafters  -=  R  = 

on  braces  —  Q  —  • 

1  to  8 

48  pounds 

49  "5  pounds. 

1   "  7 

48'6 

50-5 

"  6 

49-4 

51-9        " 

"  6 

50-6 

54- 

«  4 

52-6 

67-6 

"  3 

56-3 

64-         " 

"  2 

63-7 

76-2        " 

"   1 

81' 

101- 

To  get  the  proportion  that  the  height  bears  to  the  span,  di- 
vide the  span  by  the  height ;  then  unity  will  be  to  the  quotient 
as  the  height  is  to  the  span.  In  case  the  quotient  is  not  a 
whole  number,  the  required  value  of  R  or  Q  will  not  be  found 
in  the  above  table,  but  may  be  obtained  thus :  multiply  the 
decimal  part  of  the  quotient  by  the  difference  of  the  values  of 
R  set  opposite  the  two  proportions,  between  which  the  given 
proportion  occurs  as  an  intermediate,  and  subtract  the  product 
from  the  larger  of  the  two  said  values  of  R ;  the  remainder 
will  be  the  value  of  R  required.  The  process  is  the  same  for 
the  values  of  Q. 

Example. — A  roof  whose  span  is  60  feet,  has  a  height  of  25 
feet.  Then  60  divided  by  25  equals  2*4.  The  proportion, 
therefore,  between  the  height  and  span  is  1  to  2'4.  This  pro- 
portion is  an  intermediate  between  1  to  2  and  1  to  3.  The 
values  of  JR,  opposite  these  two,  are  63*7  and  56*3.  The  dif- 
ference between  these  values  is  7'4 ;  this  multiplied  by  0'4,  the 
decimal  portion  of  the  quotient,  equals  2'96 ;  this  subtracted 
from  63*7,  the  larger  value  of  R,  the  remainder,  60'74,  is  the 
required  ^ilue  of  R. 


TEAMING.  275 

The  values  of  R  and  Q  are  those  for  a  roof  covered  with 
slate  weighing  7  pounds  per  superficial  foot  of  the  roof  sur- 
face. "When  the  roof  covering  is  either  lighter  or  heavier,  sub- 
tract from  or  add  to  the  table  values,  the  difference  of  weight 
between  7  pounds  and  the  weight  of  the  covering  used,  and 
the  remainder,  or  sum,  will  be  the  value  of  E  or  Q  required. 

377. — Rule  LIV.  To  obtain  the  dimensions  of  braces. 
Multiply  the  value  of  Q  (Table  IX.,  Art.  376)  by  the  distance 
apart  in  feet  at  which  the  roof  trusses  are  placed,  and  by  the 
horizontal  distance  in  feet  from  a  point  half-way  to  the  next 
point  of  support  of  the  rafter  on  one  side  of  the  brace,  to  a 
corresponding  point  on  the  other  side.  The  product  will  be 
the  weight  in  pounds  sustained  at  the  head  of  the  brace.  To 
this  add  the  vertical  strain  (Art.  360)  on  the  suspension  rod 
located  at  the  head  of  the  brace,  and  make  a  vertical  line 
dropped  from  the  head  of  the  brace,  as  Fe,  Fig.  233,  equal, 
by  any  convenient  scale,  to  this  sum,  and  draw  the  parallelo- 
gram Ffeg.  Then  Ff,  measured  by  the  same  scale,  equals 
the  pressure  in  the  axis  of  the  brace  F B.  Multiply  this  pres- 
sure in  pounds  by  the  square  of  the  length  of  the  brace  in  feet, 
and  divide  the  product  by  the  breadth  of  the  brace  in  inches 
multiplied  by  the  value  of  B  (Table  II.,  Art.  293).  The  cube 
root  of  the  quotient  will  be  the  thickness  of  the  brace  in 
inches.  If  this  cube  root  should  exceed  the  breadth  of  the 
brace,  the  result  is  not  correct,  and  the  calculation  will  have  to 
be  made  anew,  taking  a  larger  dimension  for  the  breadth. 

Example. — The  brace  F  B  (Fig.  233)  is  of  white  pine,  and 
is  required  to  sustain  a  pressure  in  its  axis  of  7000  pounds 
(Art.  362).  The  length  of  the  brace  is  12  feet  and  its  breadth 
6  inches,  what  must  be  its  thickness  ?  Here  7000,  the  pressure, 
multiplied  by  144,  the  square  of  the  length,  equals  1008000. 
The  value  of  B  is  1175;  this  by  6,  the  breadth  of  the  brace, 
equals  7050.  The  product  1003000  divided  by  the  product 
7050  equals  143.  the  cube  root  of  which,  5*23,  is  the  required 


276  AMERICAN    HOUSE-CARPENTER. 

thickness  of  the  brace  in  inches.  The  brace  will  therefore  be 
5'23  by  6  inches,  or  5i  by  6. 

378. — Rule  LY.  To  obtain  the  area  of  the  abutting  sur- 
face of  the  ends  of  braces.  Divide  the  number  of  degrees 
contained  in  the  complement  of  the  angle  of  inclination  by  90, 
and  multiply  the  quotient  by  the  difference  of  the  values  of  C 
and  P,  set  opposite  the  kind  of  wood  in  the  tie  beam  or  rafter, 
in  Table  L,  Art.  292 ;  and  to  the  product  add  the  said  value 
of  P,  and  by  the  sum  divide  the  pressure  in  the  axis  of  the 
brace,  and  the  quotient  will  be  the  area  of  the  abutting  sur- 
face. 

The  complement  of  the  angle  of  inclination  referred  to  is, 
for  the  foot  of  the  brace,  the  acute  angle  contained  between 
the  brace  and  a  vertical  line ;  and  for  the  head  of  the  brace, 
the  acute  angle  contained  between  the  brace  and  a  line  per- 
pendicular to  the  rafter. 

Example. — To  find  the  abutting  surface  of  the  ends  of  the 
brace  FB  (Fig.  233).  The  complement  of  the  angle  of  incli- 
nation, for  the/botf  of  the  brace,  is  that  contained  between  the 
lines  F B  and  F  E,  and  measures  by  the  protractor,  45°.  The 
tie  beam  is  of  white  pine,  and  the  values  of  P  and  C  for  this 
wood  are  300  and  1200  respectively,  and  the  pressure  in  the 
axis  of  the  brace  is  7000  pounds.  Now  by  the  rule,  45  di- 
vided by  90  equals  0-5,  this  by  the  900,  the  difference  of1  the 
values  of  C  and  P,  equals  450  ;  to  this  add  300,  the  value  of 
P,  and  the  sum  is  750.  The  pressure  in  the  axis  of  the  brace, 
7000,  divided  by  this  750,  equals  9§,  the  required  area  of  the 
abutting  surface  at  the  foot  of  the  rafter.  The  complement  of 
the  angle  of  inclination  for  the  head  of  the  brace  is  that  con- 
tained between  the  lines  B  F  and  Fp,  and  measures  by  the 
protractor  19°.  The  rafter  being  of  white  pine,  the  values  of 
P  and  C  are  as  before.  By  the  rule,  19  divided  by  90  equals 
0'2£,  and  this  multiplied  by  900,  the  difference  of  the  values 
of  P  and  (7,  equals  190  ;  to  this  add  300,  the  value  <•?  P,  and 


FKAMIXG.  277 

the  sum  is  490.  The  pressure,  7000,  divided  by  this  490, 
equals  14'3  inches,  the  required  area  of  the  abutting  surface  at 
the  head  of  the  brace. 

879. — To  obtain  the  dimensions  of  the  tie  beam.  Tie  beams 
are  subjected  to  two  kinds  of  strain — tensile  and  transverse. 

Rule  LYI. — To  guard  against  the  tensile  strain,  multiply  the 
value  of  R  (Table  IX.,  Art.  376)  by  three  times  the  distance 
apart  at  which  the  trusses  are  placed,  and  by  the  square  of  the 
span  of  the  truss,  both  in  feet.  Divide  this  product  by  the 
value  of  r,  (Table  III.,  Art.  308)  set  opposite  the  kind  of  wood 
in  the  tie  beam,  multiplied  by  8  times  the  height  of  the  roof 
in  feet,  and  by  the  breadth  of  the  tie  beam  in  inches.  The 
quotient  will  be  the  required  depth  in  inches. 

The  result  thus  obtained  is  usually  smaller  than  that  re- 
quired to  resist  the  cross  strain  to  which  the  tie  beam  is  sub- 
jected. The  dimensions  required  to  resist  this  strain,  where 
there  is  simply  the  weight  of  the  ceiling  to  support,  may  be 
obtained  by  this  rule : 

Rule  LYII. — Multiply  the  cube  of  the  longest  unsupported 
part  of  the  tie  beam  by  400  times  the  distance  apart  at  which 
the  trusses  are  placed,  both  in  feet ;  and  divide  the  product  by 
the  breadth  of  the  tie  beam  in  inches,  multiplied  by  the  value 
of  E,  (Table  II.,  Art.  293)  set  opposite  the  kind  of  wood  in  the 
tie  beam,  and  the  cube  root  of  the  quotient  will  be  the  re- 
quired depth  of  the  tie  beam  in  inches. 

The  result  thus  obtained  may  not  be  sufficient,  in  some  cases,, 
to  resist  the  strains  to  which  the  tie  beam  is  subjected  in  the 
hands  of  the  workmen  during  the  process  of  framing. 

Rule  LVIIL — To  resist  these  strains  the  area  of  cross  sec- 
tion in  inches  should  be  at  least  equal  to  the  length  in  feet. 

JZcample. — The  tie  beam  in  Fig.  233.  For  this  case  we  have 
the  value  of  R  52*6,  the  trusses  placed  10  feet  from  centres, 
the  span  52  feet,  the  height  13  feet,  the  breadth  6  inches,  and 
the  value  of  T  2367.  Then  by  the  rule,  52-6  x  3  x  10  x  52* 


278  AMERICAN   HOUSE-CARPENTER. 

-  4266912,  and  236T  x  8  x  13  x  6  =  1477008 ;  the  formei 
product  divided  by  the  latter,  the  quotient  equals  2*9,  equals 
the  required  depth  of  the  tie  beam  in  inches.  The  other 
strains  will  require  the  depth  to  be  more.  To  resist  the  cross 
strains,  we  have  the  longest  unsupported  part  of  the  tie  beam 
8|  feet,  (this  dimension  is  frequently  greater  than  this,)  distance 
from  centres  10  feet,  and  breadth  6  inches.  Then,  by  the  rule, 
8|3  x  400  x  10  =  2603852,  and  6  x  1750  =  10500 ;  the  former 
product  divided  by  the  latter,  the  quotient  is  248,  the  cube  root 
of  which,  6*28,  equals  the  required  depth  in  inches.  The  tie 
beam  therefore  is  to  be  6  by  6*28  inches,  or  6  x  7  inches.  But 
if  not  guarded  against  severe  accidental  strains  from  careless 
handling  this  size  would  be  too  small.  It  would,  in  this  case, 
require  to  be  52  inches  area  of  cross  section,  say  6x9  inches. 

380. — To  obtain  the  diameter  of  the  suspension  rods,  when 
made  of  good  wrought  iron. 

Rule  LIX. — Divide  the  weight  or  vertical  strain,  in  pounds, 
by  4000.  The  square  root  of  the  quotient  will  be  the  required 
diameter  of  the  rod  in  inches. 

Example. — A  suspension  rod  is  required  to  sustain  16000 
pounds,  what  must  be  its  diameter  ?-  Dividing  by  4000,  the 
quotient  is  4;  the  square  root  of  which,  2,  is  the  required 
diameter. 

The  vertical  strain  on  any  rod  is  equal  to  the  weight  of  so 
much  of  the  ceiling  as  is  supported  by  the  rod,  added  to  the 
vertical  strain  caused  by  each  brace  that  is  footed  in  the  tie 
beam  at  the  rod.  The  weight  of  the  ceiling  supported  by  a 
rod,  is  equal  to  ten  times  the  distance  apart  in  feet  at  which 
the  trusses  are  placed,  multiplied  by  half  the  distance  in  feet 
between  the  two  next  points  of  support,  one  on  either  side  of 
the  rod.  The  vertical  strain  caused  by  the  braces  can  be  as 
certained  geometrically,  as  in  Art.  360. 

381. — When  the  suspension  rods  are  located  as  in  Fig.  233, 
dividing  the  span  into  equal  j  arts,  the  diameter  ol  the  roda 


FRAMING.  279 

may  be  obtained  without  the  preliminary  calculation  of  the 
strain,  as  follows  : 

Eule  LX. — For  the  first  rod  from  the  wall.  Multiply  the 
distance  apart  at  which  the  trusses  are  placed  by  the  distance 
apart  between  the  suspension  rods,  and  divide  the  product  by 
400.  The  square  root  of  the  quotient  will  be  the  required 
diameter  of  the  rod. 

Example,. — Rod  1)  G,  Fig.  233.  In  this  figure  the  rods  are 
located  at  8f  feet  apart,  and  the  distance  between  the  trusses 
is  10  feet.  Therefore,  10  x  8|  =  86^  ;  this  divided  by  400, 
the  quotient  is  0-2167,  the  square  root  of  which,  0*465,  is  the 
required  diameter.  The  diameter  may  be  half  an  inch. 

Rule  LXL— For  the  second  rod  from  the  wall.  To  the 
value  of  Q  (Table  IX.,  Art.  376)  add  20,  and  multiply  the  sum 
by  the  distance  apart  at  which  the  trusses  are  placed  and  by 
the  distance  between  the  rods,  both  in  feet,  and  divide  the  pro- 
duct by  8000.  The  square  root  of  the  quotient  will  be  the  re- 
quired diameter. 

Example.— Eod  F  E,  Fig.  233.  The  distances  apart  in  this 
case  are  as  stated  in  last  example.  The  value  of  Q  is  57'6,  and 
when  added  to  20  equals  77'6.  Therefore,  77'6  X  10  x  Sf  = 
6673-J  ;  this  divided  by  8000,  the  quotient  is  0-8341,  the  square 
root  of  which.  0'91,  is  the  required  diameter.  This  rod  may 
be  one  inch  diameter. 

Rube  LXIL— For  the  centre  rod.  To  the  value  of  Q  (Table 
IX.,  Art.  376)  add  5,  and  multiply  the  sum  by  the  distance 
apart  at  which  the  trusses  are  placed  and  by  the  distance  apart 
between  the  rods,  both  in  feet,  and  divide  the  product  by  2000. 
The  square  root  of  the  quotient  will  equal  the  required  diameter. 

Example.— Rod  C  B,  Fig.  233.  The  distances  apart  as  be- 
fore, and  the  value  of  Q  the  same.  To  Q  add  5,  and  the  sura 
is  62-'6.  Then  62'6  x  10  x  8f  =  5425£  ;  this  divided  by  2000, 
the  quotient  is  2'7126,  the  square  root  of  which,  1/647,  equal? 
the  required  diameter.  This  rod  may  be  If  inches  diameter. 


280  AMERICAN    HOUSE-CARPENTER. 

382. — For  all  wrought  iron  straps  and  bolts  the  dimensions 
may  be  found  by  this  rule. 

Rule  LXIII. — Divide  the  tensile  strain  on  the  piece,  in 
pounds,  by  5000,  and  the  quotient  will  be  the  area  of  cross 
section  of  the  required  bar  or  bolt,  in  inches. 
•  383. — Roof-beams,  jack-rafters,  and  purlins.  All  pieces  of 
timber  subject  to  cross  strains  will  sustain  safely  much  greater 
strains  when  extended  in  one  piece  over  two,  three,  or  more 
distances  between  bearings  ;  therefore  roof-beams,  jack-rafters, 
and  purlins  should,  if  possible,  be  made  in  as  long  lengths  as 
practicable ;  the  roof-beams  and  purlins  laid  on,  not  framed 
into,  the  principal  rafters,  and  extended  over  at  least  two 
spaces,  the  joints  alternating  on  the -trusses;  and  likewise  the 
jack-rafters  laid  on  the  purlins  in  long  lengths.  The  dimen- 
sions of  these  several  pieces  may  be  obtained  by  the  following 
rule : 

Rule  LXIY.— From  the  value  of  Q  (Table  IX.,  Art.  376) 
deduct  10,  and  multiply  the  remainder  by  33  times  the  distanco 
from  centres  in  feet  at  which  the  pieces  are  placed,  and  by  the 
cube  of  the  distance  between  bearings  in  feet ;  divide  the  pro- 
duct by  the  value  of  E  (Table  II.,  Art.  293)  for  the  kind  of 
wood  used  and  extract  the  square  root  of  the  quotient.  The 
square  root  of  this  square  root  will  be  the  required  depth  in 
inches.  Multiply  the  depth  thus  obtained  by  the  decimal  0'6, 
and  the  product  will  be  the  required  breadth  in  inches. 

Example. — Roof-beams  of  white  pine  placed  2  feet  from  cen- 
tres, resting  on  trusses  placed  10  feet  from  centres,  the  height 
and  the  span  of  the  roof  being  in  proportion  as  1  to  4.  In 
this  case  the  value  of  Q  is  57'6.  By  the  rule,  57'6  -  10  =  47'6, 
and  47-6  x  33  x  2  x  103  =  3141600.  This  divided  by  1750, 
the  quotient  is  1795-2,  the  square  root  of  which  is  42-37,  and  the 
square  root  of  42-37  is  6'5,  the  required  depth.  This  multi- 
plied by  0-6  equals  3-9,  the  required  breadth.  These  roof 
beams  may  therefore  be  4  by  6$  inches. 


FRAMING. 


281 


384. — Five  examples  of  roofs  are  shown  at  Figs.  234,  235, 
236,  237,  and  238.  In  Fig.  234,  a  is  an  iron  suspension  rod, 
5  5  are  braces.  In  Fig.  235, 
«,  «,  and  J  are  iron  rods,  and 
d  d,  c  <?,  are  braces.  In  Fig. 
236,  a  T)  are  iron  rods,  dd 
braces,  and  c  the  straining  U 
beam.  In  Fig.  237,  aa,lb, 


D 


are  iron  rods,  e  e,  t?  <2,  are  braces,  and  c  is  a  straining  beam. 
In  Fig.  238,  purlins  are  located  at  P  P,  &c. ;  the  inclined  beam 
that  lies  upon  them  is  the  jack-rafter;  the  post  at  the  ridge  is 
the  king  post,  the  others  are  queen  posts.  In  this  design  the  tie 
beam  is  increased  in  height  along  the  middle  by  a  strengthen- 
ing piece  (Art.  348),  for  the  purpose  of  sustaining  additional 
weight  placed  in  the  room  formed  in  the  truss. 

385. — Fig.  239  shows  a  method  of  constructing  a  truss  having 
a  luilt-rib  in  the  place  of  principal  rafters.  The  proper  form 
for  the  curve  is  that  of  a  parabola,  (Art.  127.)  This  curve, 


282 


AMERICAN   HOUSE-CARPENTEK. 


when  as  flat  as  is  described  in  the  figure,  approximates  so 
near  to  that  of  the  circle,  that  the  latter  may  be  used  in  its 
stead.  The  height,  a  5,  is  just  half  of  a  c,  the  curve  to  pass 
through  the  middle  of  the  rib. 
The  rib  is  composed  of  two  series 
of  abutting  pieces,  bolted  toge- 
ther. These  pieces  should  be  as 
long  as  the  dimensions  of  the  tim- 
ber will  admit,  in  order  that  there 
may  be  but  few  joints.  The  sus- 
pending pieces  are  in  halves, 
notched  and  bolted  to  the  tie- 
beam  and  rib,  and  a  purlin 'is 
framed  upon  the  upper  end  of 
each.  A  truss  of  this  construc- 
tion needs,  for  ordinary  roofs,  no 
<sf  |j>  diagonal  braces  between  the  sus- 
pending pieces,  but  if  extra 
strength  is  required  the  braces 
may  be  added.  The  best  place 
for  the  suspending  pieces  is  at  the 
joints  of  the  rib.  A  rib  of  this 
kind  will  be  sufficiently  strong, 
if  the  area  of  its  section  contain 
about  one-fourth  more  timber, 
than  is  required  for  that  of  a  raf- 
ter for  a  roof  of  the  same  size. 
The  proportion  of  the  depth  to 
the  thickness  should  be  about  as 
10  is  to  Y. 

386 — Some  writers  have  given  designs  for  roofs  similar  to 
Fig.  240,  having  the  tie-beam  omitted  for  the  accommodation 
of  an  arch  in  the  ceiling.  This  and  all  similar  designs  are  se- 


FRAMING.  283 

riously  objectionable,  and  should  always  be  avoided  ;  as  the 
small  height  gained  by  the  omission  of  the  tie-beam  can  never 


Fig.  240. 


compensate  for  the  powerful  lateral  strains,  which  are  exerted 
by  the  oblique  position  of  the  supports,  tending  to  separate  the 


284 


AMERICAN   HOUSE-CARPENTER. 


walls.  Where  an  arcli  is  required  in  the  ceiling,  the  best  plan 
is  to  carry  up  the  walls  as  high  as  the  top  of  the  arch.  Then, 
by  using  a  horizontal  tie-beam,  the  oblique  strains  will  be  en- 
tirely removed.  Many  a  public  building,  by  my  own  obser- 
vation, has  been  all  but  ruined  by  the  settling  of  the  roof, 
consequent  upon  a  defective  plan  in  the  formation  of  the  truss 
in  this  respect.  It  is  very  necessary,  therefore,  that  the  hori- 
zontal tie-beam  be  used,  except  where  the  walls  are  made  so 
strong  and  firm  by  buttresses,  or  other  support,  as  to  prevent 
a  possibility  of  their  separating. 


387. — Fig.  241  is  a  method  of  obtaining  the  proper  lengths  and 
bevils  for  rafters  in  a  hip-roof:  a  b  and  b  c  are  walls  at  the  angle 
of  the  building ;  b  e  is  the  seat  of  the  hip-rafter  and  gf  of  a 
jack  or  cripple  rafter.  Draw  e  A,  at  right  angles  to  b  e,  and  make 
it  equal  to  the  rise  of  the  roof;  join  b  and  A,  and  h  b  will  be  the 
length  of  the  hip-rafter.  Through  <?,  draw  d  i,  at  right  angles  to 
be /  upon  J,  with  the  radius,  bh,  describe  the  arc,  hi,  cutting di 
in  .i  /  join  b  and  i,  and  extend  gf  to  meet  biinj  /  then  gj  will 


FRAMING. 


285 


be  the  length  of  the  jack-rafter.  The  length  of  each  jack-rafter  is 
found  in  the  same  manner — by  extending  its  seat  to  cut  the  line, 
b  i.  From/,  draw  fk,  at  right  angles  tofg,  also/Z,  at  right 
angles  to  be;  makef/c  equal  to  fl  by  the  arc,  I  k,  or  make  g  k 
equal  to  g  j  by  the  arc,  j  k  ;  then  the  angle  at  j  will  be  the  top- 
bevil  of  the  jack-rafters,  and  the  one  at  k  will  be  the  down-bevil* 
388. —  To  find  the  backing  of  the  hip-rafter.  At  any  con- 
venient place  in  b  e,  (Fig.  241,)  as  o,  draw  m  n,  at  right  angles  to 
be;  from  o,  tangical  to  b  h,  describe  a  semi-circle,  cutting  b  e  in 
s  ;  join  m  and  5  and  n  and  5  ;  then  these  lines  will  form  at  5  the 
proper  angle  for  beviling  the  top  of  the  hip-rafter. 

DOMES.! 


ij — or 


Fig.  248. 

•  The  lengths  and  bevils  of  rafters  for  roof-valleys  can  also  be  found  by  tie  abore 
process  f  See  also  Art.  237. 


286 


AMERICAN    HOUSE-CARPENTER. 


389.— The  most  usual  form  for  domes  is  that  of  the  sphere,  the 
base  being  circular.  When  the  interior  dome  does  r  ot  rise  toe 
high,  a  horizontal  tie  may  be  thrown  across,  by  which  any  de- 
gree of  strength  required  may  be  obtained.  Fig.  242  shows  a 
section,  and  Fig.  243  the  plan,  of  a  dome  of  this  kind,  a  b  being 
the  tie-beam  in  both.  Two  trusses  of  this  kind,  (Fig.  242,)  pa- 
rallel to  each  other,  are  to  be  placed  one  on  each  side  of  the  open- 
ing in  the  top  of  the  dome.  Upon  these  the  whole  framework  is  to 
depend  for  support,  and  their  strength  must  be  calculated  accord- 
ingly. (See  the  first  part  of  this  section,  and  Art.  356.)  If  the 
dome  is  large  and  of  importance,  two  other  trusses  may  be  intro- 
duced at  right  angles  to  the  foregoing,  the  tie-beams  being  pre- 
served in  one  continuous  length  by  framing  them  high  enough  to 
pass  over  the  others. 


Fig.  244 


390.—- When  the  interior  dome  rises  too  high  to  admit  of  a  leve, 


FRAMING.      '  287 

tie-beam,  the  framing  may  be  composed  of  a  succession  of  ribs 
standing  upon  a  continuous  circular  curb  of  timber,  as  seen  at 
Fig.  244  and  245; — the  latter  being  a  plan  and  the  former  a  sec- 
tion. This  curb  must  be  well  secured,  as  it  serves  in  the  place 
of  a  tie-beam  to  resist  the  lateral  thrust  of  the  ribs.  In  small 
domes.  th°se  ribs  may  be  easily  cut  from  wide  plank ;  but,  where 
an  extensive  structure  is  required,  they  must  be  built  in  two 
thicknesses  so  as  to  break  joints,  in  the  same  manner  as  is  descri- 
bed for  a  roof  at  Art.  385.  They  should  be  placed  at  about  two 
feet  apart  at  the  base,  and  strutted  as  at  a  in  Fig.  244. 

391. — The  scantling  of  each  thickness  of  the  rib  may  be  as 
follows : 

For  domes  of  24  feet  diameter,  1x8   inches. 

"         "         36  "  lixlO     " 

','         '        60  "  2x13     " 

"        «        90  "          2^x13     " 

"        "      108  "  3x13    " 

392. — Although  the  outer  and  the  inner  surfaces  of  a  dome 
may  be  finished  to  any  curve  that  may  be  desired,  yet  the  framing 
should  be  constructed  of  such  a  form,  as  to  insure  that  the  curve 
of  equilibrium  will  pass  through  the  middle  of  the  depth  of  the 
framing.  The  nature  of  this  curve  is  such  that,  if  an  arch  or 
dome  be  constructed  in  accordance  with  it,  no  one  part  of  the 
structure  will  be  less  capable  than  another  of  resisting  the  strains 
and  pressures  to  which  the  whole  fabric  may  be  exposed.  The 
curve  of  equilibrium  for  an  arched  vault  or  a  roof,  where  the  load 
is  equally  diffused  over  the  whole  surface,  is  that  of  a  parabola, 
(Art.  127  ;)  for  a  dome,  having  no  lantern,  tower  or  cupola  above 
it,  a.  cubic  parabola,  (Fig.  246  :)  and  for  one  having  a  tower,  &c., 
above  it,  a  curve  approaching  that  of  an  hyperbola  must  be  adopted, 
as  the  greatest  strength  is  required  at  its  upper  parts.  If  the 
curve  of  a  dome  be  circular,  (as  in  the  vertical  section,  Fig.  244,) 
the  pressure  will  have  a  tendency  to  burst  the  dome  outwards  at 
about  one-third  of  its  height.  Therefore,  when  this  form  is  used 


288 


AMERICAN    HOUSE  CARPENTER. 


in  the  construction  of  an  extensive  dome,  an  iron  band  should  be 
placed  around  the  framework  at  that  height ;  and  whatever  maj 
be  the  form  of  the  curve,  a  bond  or  tie  of  some  kind  is  necessary 
around  or  across  the  base. 

If  the  framing  be  of  a  form  less  convex  than  the  curve  of 
equilibrium,  the  weight  will  have  a  tendency  to  crush  the  ribs  in- 
wards, but  this  pressure  may  be  effectually  overcome  by  strutting 
between  the  ribs  ;  and  hence  it  is  important  that  the  struts  be  so 
placed  as  to  form  c  ntinuous  horizontal  circles. 


Fig.  246. 

393. —  To  describe  a  cubic  parabola.  Let  a  6,  (Fig.  246,)  be 
the  base  and  b  c  the  height.  Bisect  a  b  at  rf,.and  divide  a  d  into 
100  equal  parts;  of  these  give  d  e  26,  ef  18J,/  g  14  J,  g  h  12£, 
h  i  10$,  ij  9i,  and  the  balance,  8|,  to  j  a;  divide  b  c  into  8  equal 
parts,  and,  from  the  points  of  division,  draw  lines  parallel  to  a  b, 
to  meet  perpendiculars  from  the  several  points  of  division  in  a  b, 
at  the  points,  o,  o,  o,  &c.  Then  a  curve  traced  through  these 
points  will  be  the  one  required. 

394. — Small  domes  to  light  stairways,  &c.,  are  frequently  made 
elliptical  in  both  plan  and  section ;  and  as  no  two  of  the  ribs  in 
one  quarter  of  the  dome  are  alike  in  form,  a  method  for  obtaining 
the  curves  is  necessary. 

395. —  To  find  the  curves  for  the  ribs  of  an  elliptical  dome 
Let  a  b  c  d,  (Fig.  247",)  be  the  plan  of  a  dome,  and  e  f  the  seat 


FRAMING. 


289 


Fig.  247. 


01  one  of  the  ribs.  Then  take  e/for  the  transverse  axis  ana 
twice  the  rise,  o  g,  of  the  dome  for  the  conjugate,  and  describe 
(according  to  Art.  115,  116,  &c.,)  the  semi-ellipse,  e  gf,  which 
will  be  the  curve  required  for  the  rib,  e  g  f.  The  other  ribs  are 
found  in  the  same  manner. 


Fig.  248. 

396. —  To  find  the  shape  of  the  covering  for  a  spherical 
dome.  Let  A,  (Fig.  248,)  be  the  plan  and  B  the  section  of  a 
given  dome.  From  a,  draw  a  c,  at  right  angles  to  a  b  ;  find  the 
stretch-out,  (Art.  92,)  of  o  &,  and  make  d  c  equal  to  it ;  divide  the 
arc,  o  b,  and  the  line,  d  c,  each  into  a  like  number  of  equal  parts, 

37 


290 


AMERICAN    HOUSE-CARPENTER. 


as  5,  (a  large  number  will  insure  greater  accuracy  than  a  small 
one  ;)  upon  c,  through  the  several  points  of  division  in  c  d,  describe 
the  arcs,  o  d  o,  1  e  1,  2f  2,  &c. ;  make  d  o  equal  to  half  the  width 
of  one  of  the  boards,  and  draw  o  s,  parallel  to  a  c  ;  join  5  and  a, 
and  from  the  points  of  division  in  the  arc,  o  b,  drop  perpendicu- 
lars, meeting  a  5  in  ij  k  I  ;  from  these  points,  draw  i  4,  j  3,  &c., 
parallel  to  a  c;  make  d  o,  e  I,  &c.,  on  the  lower  side  of  a  c,  equal 
to  d  o,  e  1,  &c.,  on  the  upper  side  ;  trace  a  curve  through  the 
points,  o,  1,  2,  3,  4,  c,  on  each  side  of  c?  c  ;.  then  o  c  o  will  be 
the  proper  shape  for  the  board.  By  dividing  the  circumference  of 
the  base,  A,  into  equal  parts,  and  making  the  bottom,  o  d  o,  of  the 
board  of  a  size  equal  to  one  of  those  parts,  every  board  may  be 
made  of  the  same  size.  In  the  same  manner  as  the  above,  the 
"hape  of  the  covering  for  sections  of  another  form  may  be  found, 
such  as  an  ogee,  cove,  &c. 


397. —  To  find  the  curve  of  the  boards  when  laid  in  horizon- 
tal courses.  Let  ABC,  (Fig.  249,)  be  the  section  of  a  given 
dome,  and  D  B  its  axis.  Divide  B  C  into  as  many  parts  as 
there  are  to  be  courses  of  boards,  in  the  points,  1,  2,  3,  &e.  i 
through  1  and  2.  draw  a  line  to  meet  the  axis  extended  at  a  . 
then  a  will  be  the  centre  for  describing  the  edges  of  the  board.  E. 
Through  3  and  2,  draw  36;  then  b  will  be  the  centre  for  describing 
F.  Through  4  and  3,  draw  4  d;  then  d  will  be  the  centre  for  G, 
B  is  the  centre  for  the  arc,  1  o.  If  this  method  is  taker,  ta  find 


FRAMING. 


291 


the  centres  for  the  boards  at  the  base  of  the  dome,  they  would 
occur  so  distant  as  to  make  it  impracticable  :  the  following  metl  od 
is  preferable  for  this  purpose.  G  being  the  last  board  obtained  by 
the  above  method,  extend  the  curve  of  its  inner  edge  until  it 
merts  the  axis,  D  B,  in  e  ;  from  3,  through  e,  draw  3/,  meeting 
the  arc,  A  JB,  in/;  join /and  4, /and  5  and /and  6,  cutting  the 
axis,  D  B,  in  s,  n  and  m ;  from  4,  5  and  6,  draw  lines  parallel  to 
A  C  and  cutting  the  axis  in  c,p  and  r  ;  make  c  4,  (Fig.  250,) 


Fig.  250. 


equal  to  c  4  in  the  previous  figure,  and  c  s  equal  to  c  s  also  in  the 
previous  figure ;  then  describe  the  inner  edge  of  the  board,  H, 
according  to  Art.  87  :  the  outer  edge  can  be  obtained  by  gauging 
from  the  inner  edge.  In  like  manner  proceed  to  obtain  the  next 
board — taking  p  5  for  half  the  chord  and  p  n  for  the  height  of  the 
segment.  Should  the  segment  be  too  large  to  be  described 
easily,  reduce  it  by  finding  intermediate  points  in  the  curve,  as  at 
Art.  86. 


398. —  To  find  the  shape  of  the  angle-rib  for  a  polygonal 
dcme.    Let  A  G  H,  (Fig.  251,)  be  the  plan  of  a  given  dome,  and 


292 


AMERICAN    HOUSE-CARPENTER. 


C  D  a  vertical  section  taken  at  the  line,  ef.  From  1,  2,  3,  &c. 
in  the  arc,  C  D,  draw  ordinates,  parallel  to  A  Z>,  to  meet/  G  , 
from  the  points  of  intersection  on  /  G,  draw  ordinates  at  right- 
angles  to/  G;  make  s-l  equal  to  o  1,  5  2  equal  to  o  2,  &c. ;  then 
G  f  B,  obtained  in  this  way,  will  be  the  angle-rib  required.  The 
best  position  for  the  sheathing-boards  for  a  dome  of  this  kind  is 
horizontal,  but  if  they  are  required  to  be  bent  from  the  base  tc 
the  vertex,  their  shape  may  be  found  in  a  similar  manner  to  that 
shown  at  Fig.  248. 

BRIDGES. 

399. — Various  plans  have  been  adopted  for  the  construction  of 
bridges,  of  which  perhaps  the  following  are  the  most  useful. 
Fig.  252  shows  a  method  of  constructing  wooden  bridges,  where 
the  banks  of  the  river  are  high  enough  to  permit  the  use  of  the 
tie-beam,  a  b.  The  upright  pieces,  c  d,  are  notched  and  bolted 
on  in  pairs,  for  the  support  of  the  tie-beam.  A  bridge  of  this 
construction  exerts  no  lateral  pressure  upon  the  abutments.  This 
method  may  be  employed  even  where  the  banks  of  the  river  are 
low,  by  letting  the  timbers  for  the  roadway  rest  immediately  upon 
the  tie-beam.  In  this  case,  the  framework  above  will  serve  the 
purpose  of  a  railing. 


Fig.  252. 


400. — Fig.  253  exhibits  a  wooden  bridge  without  a  tie-beam. 
Where  staunch  buttresses  can  be  obtained,  this  method  may  be 
recommended ;  but  if  there  is  any  doubt  of  their  stability,  it 


FRAMING. 


293 


Fig.  253. 


should  not  be  attempted,  as  it  is  evident  that  such  a  system  o( 
framing  is  capable  of  a  tremendous  lateral  thrust. 


Fig-  254 


401. — Fig.  254  represents  a  wooden  bridge  in  which  a  built-i  ib, 
(see  Art.  385,)  is  introduced  as  a  chief  support.  The  curve  ot 
equilibrium  will  not  differ  much  from  that  of  a  parabola :  this, 
therefore,  may  be  used — especially  if  the  rib  is  made  gradually  a 
little  stronger  as  it  approaches  the  buttresses.  As  it  is  desirable 
that  a  bridge  be  kept  low,  the  following  table  is  given  to  show  the 
least  rise  that  may  be  given  to  the  rib. 


Span  in  feet|  Least  ri»e  in  feet 

Span  in  feet 

Least  rise  in  feet  ||  Span  in  feet 

Least  rise  in  feet. 

30 

0-5 

120 

7 

280 

24 

40 

0-8 

140 

8 

300 

23 

50 

1-4 

160 

10 

320 

32 

60 

2 

180 

11 

350 

39 

70 

2£ 

200 

12 

380 

47 

80 

3 

220 

14 

400 

53 

90 

4 

240 

17 

100 

5 

260 

20 

The  rise  should  never  be  made  less  than  this,  but  in  all  cases 


294 


AMERICAN    HOUSE-CARPENTER. 


greater  if  practicable ;  as  a  small  rise  requires  a  greater  quantity 
of  timber  to  make  the  bridge  equally  strong.  The  greatest  uni- 
form weight  with  which  a  bridge  is  likely  to  be  loaded  is,  prola- 
bly,  that  of  a  dense  crowd  of  people.  This  may  be  estimated  at 
66  pounds  per  square  foot,  and  the  framing  and  gravelled  road- 
way at  234  pounds  more  ;  which  amounts  to  300  pounds  on  a 
square  foot.  The  following  rule,  based  upon  this  estimate,  may 
be  useful  in  determining  the  area  of  the  ribs.  Rule  LXV. — 
Multiply  the  width  of  the  bridge  by  the  square  of  half  the  span, 
both  in  feet ;  and  divide  this  product  by  the  rise  in  feet,  multi- 
plied by. the  number  of  ribs;  the  quotient,  multiplied  by  the 
decimal,  O'OOll,  will  give  the  area  of  each  rib  in  feet.  When 
the  roadway  is  only  planked,  use  the  decimal,  0*0007,  instead  of 
0-0011.  Example.— What  should  be  the  area  of  the  ribs  for  a 
bridge  of  200  feet  span,  to  rise  15  feet,  and  be  30  feet  wide,  with 
3  curved  ribs  ?  The  half  of  the  span  is  100  and  its  square  is 
10,000;  this,  multiplied  by  30,  gives  300,000,  and  15,  multi- 
plied by  3,  gives  45 ;  then  300,000,  divided  by  45,  gives  6666f, 
which,  multiplied  by  O'OOll,  gives  7'333  feet,  or  1056  inches  for 
the  area  of  each  rib.  Such  a  rib  may  be  24  inches  thick  by  44 
inches  deep,  and  composed  of  6  pieces,  2  in  width  and  3  in  depth. 


Fig.  256. 


402. — The  above  rule  gives  the  area  of  a  rib,  that  would  be  re- 
quisite to  support  tho  greatest  possible  uniform  load.  But  in 
large  bridges,  a  variable  load,  such  as  a  heavy  wagon,  is  capable 
of  exerting  much  greater  strains ;  in  such  cases,  therefore,  the 
rib  should  be  made  larger.  The  greatest  concentrated  load  a 


FRAMING,  295 

bridge  will  be  likely  to  encounter,  may  be  estimated  at  from  alout 
20  to  50  thousand  pounds,  according  to  the  size  of  the  bridge. 
This  is  capable  of  exerting  the  greatest  strain,  when  placed  at 
about  one-third  of  the  span  from  one  of  the  abutments,  as  at  b 
(Fig.  255.)  The  weakest  point  of  the  segment,  b  g  c,  is  at  gt 
the  most  distant  point  from  the  chord  line.  The  pressure  exerted 
at  b  by  the  above  weight,  may  be  considered  to  be  in  the  direction 
of  the  chord  lines,  b  a  and  be;  then,  by  constructing  the  paral- 
lelogivn  of  forces,  e  b  f  d,  according  to  Art.  258,  b  f  will  show 
the  pressure  in  the  direction,  b  c.  Then  the  scantling  for  the  rib 
may  be  found  by  the  following  rule. 

Rule  LXV1. — Multiply  the  pressure  in  pounds  in  the  direc- 
tion b  c,  bj  the  distance  g  h,  and  by  the  square  of  the  distance 
b  c,  both  in  feet;  and  divide  the  product  by  the  united  breadth 
in  inches  of  the  several  ribs,  multiplied  by  the  value  of  B 
(Table  II.,  Art.  293)  for  the  kind  of  wood  used;  and  the  cube 
root  of  the  quotient  will  be  the  required  depth  of  the  rib  in 
inches. 

Example. — A  bridge  is  to  have  three  white  pine  ribs  each 
20  inches  wide ;  the  pressure  in  the  direction  b  c,  (Fig.  255) 
is -equal  to  60,000  pounds,  the  distance  b  c  equals  60  feet,  and 
the  distance  g  h  equals  10  feet.  What  must  be  the  depth  of 
the  ribs,  the  value  of  B  (Table  II.)  being  for  white  pine  1175  ? 
Here,  by  the  rule,  60,000  X  10  x  60'  =  2,160,000,000.  Then 
1175  X  3  x  20  =  70,500.  The  former  product  divided  by  the 
latter  equals  30,638,  the  cube  root  of  which,  31-29,  equals 
the  required  depth  in  inches.  The  ribs  are,  therefore,  to  be  20 
by  31  £  inches. 

403. — In  constructing  these  ribs,  if  the  span  be  not  over  50 
feet,  each  rib  may  be  made  in  two  or  three  thicknesses  of  timber, 
(three  thicknesses  is  preferable,)  of  convenient  lengths  bolted 
together ;  but,  in  larger  spans,  where  the  rib  will  be  such  as  to 
render  it  difficult  to  procure  timber  of  sufficient  breadth,  they 
may  be  constructed  by  bending  the  pieces  to  the  proper  curve 


206 


AMERICAN    HOUSE-CARPENTER. 


ind  bolting  them  together.  In  this  case,  where  timber  of  sum 
cieiit  length  to  span  the  opening  cannot  be  obtained  and  scarfing 
is  necessary,  such  joints  must  be  made  as  will  resist  both  tension 
und  compression,  (see  Fig.  264 )  To  ascertain  the  greatest  depth 
for  the  pieces  which  compose  the  rib,  so  that  the  process  of  bend 
ing  may  not  injure  their  elasticity,  multiply  the  radius  of  curvature 
in  feet  by  the  decimal,  0'05,  and  the  product  will  oe  the  depth  ir, 
inches.  Example. — Suppose  the  curve  of  the  rib  to  be  described 
with  a  radius  of  100  feet,  then  what  should  be  the  depth  ?  The 
radius  in  feet,  100,  multiplied  by  0'05,  gives  a  product  of  5  inches. 
White  pine  or  oak  timber,  5  inches  thick,  would  freely  bend  to 
the  above  curve  ;  and,  if  the  required  depth  of  such  a  rib  be  21 
inches,  it  would  have  to  be  composed  of  at  least  4  pieces.  Pitch 
pine  is  not  quite  so  elastic  as  white  pine  or  oak — its  thickness 
may  be  found  by  using  the  decimal,  0'046,  instead  of  0*05. 


404. — When  the  span  is  over  250  feet;  a  framed  rib,  formed  ac> 
in  Fig.  256,  would  be  preferable  to  the  foregoing.  Of  this,  the 
upper  and  the  lower  edges  are  formed  as  just  described,  by  bend- 
ing the  timber  to  the  proper  curve.  The  pieces  that  tend  to  the 
centre  of  the  curve,  called  radials,  are  notched  and  bolted  on  in 
pairs,  and  the  cross-braces  are  halved  together  in  the  middle,  and 
abut  end  to  end  between  the  radials.  The  distance  between  the 
ribs  of  a  bridge  should  not  exceed  at  out  8  feet.  The  roadway 


FRAMING.  297 

should  be  supported  by  vertical  standards  bolted  to  the  ribs  ai 
about  every  10  to  15  feet.  At  the  place  where  they  rest  on  the 
nbs,  a  double,  horizontal  tie  should  be  notched  and  bolted  on  the 
back  of  the  ribs,  and  also  another  on  the  under  side ;  and  diago- 
na1  braces  should  be  framed  between  the  standards,  over  the  space 
between  the  ribs,  to  prevent  lateral  motion.  The  timbers  for  the 
roadway  may  be  as  light  as  their  situation  will  admit,  as  all  use- 
less timber  is  only  an  unnecessary  load  upon  the  arch. 

405. — It  is  found  that  if  a  roadway  be  18  feet  wide,  two  car- 
riages can  pass  one  another  without  inconvenience.  Its  width 
therefore,  should  be  either  9,  18,  27  or  36  feet,  according  to  the 
amount  of  travel.  The  width  of  the  foot-path  should  be  2  feet 
for  every  person.  When  a  stream  of  water  has  a  rapid  current, 
as  few  piers  as  practicable  should  be  allowed  to  obstruct  its 
course  ;  otherwise  the  bridge  will  be  liable  to  be  swept  away  by 
freshets.  When  the  span  is  not  over  300  feet,  and  the  banks  of 
the  river  are  of  sufficient  height  to  admit  of  it,  only  one  arch 
should  be  employed.  The  rise  of  the  arch  is  limited  by  the  form 
of  the  roadway,  and  by  the  height  of  the  banks  of  the  river 
(See  Art.  401.)  The  rise  of  the  roadway  should  not  exceed  one 
in  24  feet,  but,  as  the  framing  settles  about  one  in  72,  the  roadway 
should  be  framed  to  rise  one  in  18,  that  it  may  be  one  in  24  after 
settling.  The  commencement  of  the  arch  at  the  abutments — the 
spring,  as  it  is  termed,  should  not  be  below  high-water  mark  : 
and  the  bridge  should  be  placed  at  right  angles  with  the  course  of 
the  current. 

406. — The  best  material  for  the  abutments  and  piers  of  a 
bridge,  is  stone  ;  and,  if  possible,  stone  should  be  prDcured  for  the 
purpose.  The  following  rule  is  to  determine  the  extent  of  the 
abutments,  they  being  rectangular,  and  built  with  stone  weighing 
120  Ibs.  to  a  cubic-foot.  Rule  LXYIL— Multiply  the  square 
of  the  height  of  the  abutment  by  160,  and  divide  this  product  by 
the  weight  of  a  square  foot  of  the  arch,  and  by  the  rise  of  the  arch ; 
add  unity  to  the  quotient,  and  extract  the  square-root.  Diminish 
the  square-root  by  unity, and  multiply  the  root,  so  diminished,  by 


298  AMERICAN    HOUSE-CARPENTER. 

half  the  span  of  the  arch,  and  by  the  weight  of  a  square-foot  of 
the  arch.  Divide  the  last  product  by  120  times  the  height  of  the 
abutment,  and  the  quotient  will  be  the  thickness  of  the  abutment. 
Example. — Let  the  height  of  the  abutment  from  the  base  to  the 
springing  of  the  arch  be  20  feet,  half  the  span  100  feet,  the  weight 
of  a  square  foot  of  the  arch,  including  the  greatest  possible  load 
upon  it,  300  pounds,  and  the  rise  of  the  arch  18  feet — what  should 
be  its  thickness  ?  The  square  of  the  height  of  the  abutment, 
400,  multiplied  by  160,  gives  64,000,  and  300  by  18,  gives  5400 ; 
64,000,  divided  by  5400,  gives  a  quotient  of  11-852,  one  added  to 
this  makes  12-852,  the  square-root  of  which  is  3*6  ;  this,  less  one, 
is  2-6 ;  this,  multiplied  by  100,  gives  260,  and  this  again  by  300, 
gives  78,000  ;  this,  divided  by  120  times  the  height  of  the  abut- 
ment, 2400,  gives  32  feet  6  inches,  the  thickness  required. 

The  dimensions  of  a  pier  will  be  found  by  the  same  rule. 
For,  although  the  thrust  of  an  arch  may  be  balanced  by  an  ad- 
joining arch,  when  the  bridge  is  finished,  and  while  it  remains 
uninjured  ;  yet,  during  the  erection,  and  in  the  event  of  one  arch 
being  destroyed,  the  pier  should  be  capable  of  sustaining  the  en- 
tire thrust  of  the  other. 

407. — Piers  are  sometimes  constructed  of  timber,  their  princi- 
pal strength  depending  on  piles  driven  into  the  earth,  but  such 
piers  should  never  be  adopted  where  it  is  possible  to  avoid  them ; 
for,  being  alternately  wet  and  dry,  they  decay  much  socner  than 
the  upper  parts  of  the  bridge.  Spruce  and  elm  are  considered 
good  for  -piles.  Where  the  height  from  the  bottom  of  the 
river  to  the  roadway  is  great,  it  is  a  good  plan  to  cut  them  off  at 
a  little  below  low-water  mark,  cap  them  with  a  horizontal  tie. 
and  upon  this  erect  the  posts  for  the  support  of  the  roadway. 
This  method  cuts  off  the  part  that  is  continually  wet  from  that 
which  is  only  occasionally  so,  and  thus  affords  an  opportunity  for 
replacing  the  upper  part.  The  pieces  which  are  immersed  will 
last  a  great  length  of  time,  especially  when  of  elm ;  for  it  is  a 
well-established  fact,  that  timber  is  less  durable  when  subject  tc 


FRAMING. 


299 


ilterm  te  dryness  and  moisture,  than  when  it  is  either  continually 
wet  or  continually  dry.  It  has  been  ascertained  that  the  piles 
jrider  London  bridge,  after  having  been  driven  about  600  years, 
veie  not  materially  decayed.  These  piles  are  chiefly  of  elm,  and 
vholly  immersed. 


Fig.  26T. 

408. — Centres  for  stone  bridges.  Fig.  257  is  a  design  fov  a 
centre  for  a  stone  bridge  where  intermediate  supports,  as  piles 
driven  into  the  bed  of  the  river,  are  practicable.  Its  timbers  are 
so  distributed  as  to  sustain  the  weight  of  the  arch-stones  as  they 
are  being  laid,  without  destroying  the  original  form  of  the  centre  ; 
and  also  to  prevent  its  destruction  or  settlement,  should  any  of  the 
piles  be  swept  away.  The  most  usual  error  in  badly-constructed 
centres  is,  that  the  timbers  are  disposed  so  as  to  cause  the  framing 
to  rise  at  the  crown,  during  the  laying  of  the  arch-stones  up  the 
sides.  To  n  medy  this  evil,  some  have  loaded  the  crown  with 
heavy  stones  ;  but  a  centre  properly  constructed  will  need  no 
such  precaution. 

Experiments  have  shown  that  an  arch-stone  does  not  .press 
apon  the  centring,  until  its  bed  is  inclined  to  the  horizon  at  an 
angle  of  from  30  to  45  degrees;  according  to  the  hardness  of  the 
stone,  and  whether  it  is  laid  in  mortar  or  not.  For  general  pur- 
poses, the  point  at  which  the  pressure  commences,  may  be  con- 
sidered to  be  at  that  joint  which  forms  an  angle  of  32  degrees 
with  the  horizon.  At  this  point,  the  pressure  is  inconsiderable. 


300  AMERICAN   HOU8E-CARPENTEB. 

but  gradually  increases  towards  the  crown.  The  following 
table  gives  the  portion  of  the  weight  of  the  arch  stones  that 
presses  upon  the  framing  at  the  various  angles  of  inclination 
formed  by  the-  bed  of  the  stone  with  the  horizon.  The  press- 
ure perpendicular  to  the  curve  is  equal  to  the  weight  of  the 
arch  stone  multiplied  by  the  decimal 

•0,  when  the  angle  of  inclination  is  32  degrees. 

•04  "  "  "  "  "  «  34      " 

•08  "  "  "  "  '•  «  36  " 

•12  "  "  «  «  «  38  " 

•17  "  "  «  «  "  '<  40  " 

•21  "  "  «  «  «  «  42  " 

•25  "  «  «  «  «  «  44  " 

•29  "  "  "  «  «  «  46  " 

•33  "  «  "  "  «  »  48  " 

•37  "  '<  "  "  "  «  60  " 

*4  **  "  52  " 

•44  "  "  "  «  «  "  64  " 

•48  "  "  .    "  "  «  «  56  " 

•52  "  "  "  "  "  "68  " 

•54  "  «  «  "  «  «  60  « 

From  this  it  is  seen  that  at  the  inclination  of  M  degrees  the 
pressure  equals  one-quarter  the  weight  of  the  stone ;  at  57  de- 
grees, half  the  weight ;  and  when  a  vertical 
line,  as  a  5,  (Fig.  258,)  passing  through  the 
centre  of  gravity  of  the  arch-stone,  does  not 
fall  within  its  bed,  c  d,  the  pressure  may  be 
considered  equal  to  the  whole  weight  of  the 
stone.  This  will  be  the  case  at  about  60  de- 
grees, when  the  depth  of  the  stone  is  double  its  breadth.  The 
direction  of  these  pressures  is  considered  in  a  line  with  the  ra- 
dius of  the  curve.  The  weight  upon  a  centre  being  known, 
the  pressure  may  be  estimated  and  the  timber  calculated  ac- 
cordingly. But  it  must  be  remembered  that  the  whole  weight 
is  never  placed  upon  the  framing  at  once — as  seems  to  have 


FRAMING. 


301 


been  the  idea  had  in  view  by  the  designers  of  some  centres. 
In  building  the  arch,  it  «,hould  be  commenced  at  each  buttress 
at  the  same  time,  (as  is  generally  the  case,)  and  each  side 
should  progress  equally  towards  the  crown.  In  designing  the 
framing,  the  effect  produced  by  each  successive  layer  of  stone 
should  be  considered.  The  pressure  of  the  stones  upon  one 
side  should,  by  the  arrangement  of  the  struts,  be  counterpoised 
by  that  of  the  stones  upon  the  other  side. 

409. — Over  a  river  whose  stream  is  rapid,  or  where  it  is  ne- 
cessary to  preserve  an  uninterrupted  passage  for  the  purposes 
of  navigation,  the  centre  must  be  constructed  without  interme- 
diate supports,  and  without  a  continued  horizontal  tie  at  the 


Fig.  259. 

blase ;  such  a  centre  is  shown  at  Fig.  259.  In  laying  the  stones 
fr*om  the  base  up  to  a  and  c,  the  pieces,  b  d  and  b  d,  act  as  ties 
to  prevent  any  rising  at  5.  After  this,  while  the  stones  are 
being  laid  from  a  and  from  c  to  5,  they  act  as  struts :  the  piece, 
fg,  is  added  for  additional  security.  Upon  this  plan,  with 
some  variation  to  suit  circumstances,  centres  may  be  con- 
structed for  any  span  usual  in  stone-bridge  building. 

410. — In  bridge  centres,  the  principal  timbers  should  abut, 
and  not  be  intercepted  by  a  suspension  or  radial  piece  between. 
These  should  be  in  halves,  notched  on  each  side  and  bolted. 
The  timbers  should,  intersect  as  little  as  possible,  for  the  more 


302  AMERICAN   UOUSE-CARPENTEK. 

joints  the  greater  is  the  settling  ;  and  halving  them  together  is 
a  bad  practice,  as  it  destroys  nearly  one-half  the  strength  of 
the  timber.  Ties  should  be  introduced  across,  especially  where 
many  timbers  meet ;  and  as  the  centre  is  to  serve  but  a  tem- 
porary purpose,  the  whole  should  be  designed  with  a  view  to 
employ  the  timber  afterwards  for  other  uses.  For  this  reason, 
all  unnecessary  cutting  should  be  avoided. 

411. — Centres  should  be  sufficiently  strong  to  preserve  a 
staunch  and  steady  form  during  the  whole  process  of  building ; 
for  any  shaking  or  trembling  will  have  a  tendency  to  prevent 
the  mortar  or  cement  from  setting.  For  this  purpose,  also, 
the  centre  should  be  lowered  a  trifle  immediately  after  the 
key-stone  is  laid,  in  order  that  the  stones  may  take  their  bear- 
ing before  the  mortar  is  set;  otherwise  the  joints  will  open  on 
the  under  side.  The  trusses,  in  centring,  are  placed  at  the 
distance  of  from  4  to  6  feet  apart,  according  to  their  strength 
and  the  weight  of  the  arch.  Between  every  two  trusses,  diago- 
nal braces  should  be  introduced  to  prevent  lateral  motion. 

412. — In  order  that  the  centre  may  be  easily  lowered,  the 
frames,  or  trusses,  should  be  placed  upon  wedge-formed  sills  ; 
as  is  shown  at  d,  (Fig.  259.)  These  are  contrived  so  as  to  ad- 
mit of  the  settling  of  the  frame  by  driving  the  wedge,  <?,  with 
a  maul,  or,  in  large  centres,  with  a  piece  of  timber  mounted  as 
a  battering-ram.  The  operation  of  lowering  a  centre  shouM 
be  very  slowly  performed,  in  order  that  the  parts  of  the  arch 
may  take  their  bearing  uniformly.  The  wedge  pieces,  instead 
of  being  placed  parallel  with  the  truss,  are  sometimes  made 
sufficiently  long  and  laid  through  the  arch,  in  a  direction  at 
right  angles  to  that  shown  at  Fig.  259.  This  method  obviates 
the  necessity  of  stationing  men  beneath  the  arch  during  the 
process  of  lowering  ;  and  was  originally  adopted  with  success 
soon  after  the  occurrence  of  an  accident,  in  lowering  a  centre, 
by  which  nine  men  were  killed. 

413. — To  give  some  idea  of  the  manner  of  estimating  the  pres 


TEAMING.  303 

snres,  in  order  to  select  timber  of  the  proper  scantling,  calculate 
(Art.  408)  the  pressure  of  the  arch-stones  from  i  to  5,  (Fig.  259,) 
and  suppose  half  this  pressure  concentrated  at  «,  and  acting  in 
the  direction  af.  Then,  by  the  parallelogram  of  forces,  (Art. 
258,)  the  strain  in  the  several  pieces  composing  the  frame, 
b  d  a,  may  be  computed.  Again,  calculate  the  pressure  of  that 
portion  of  the  arch  included  between  a  and  c,  and  consider 
half  of  it  collected  at  5,  and  acting  in  a  vertical  direction ; 
then,  by  the  parallelogram  of  forces,  the  pressure  on  the  beams, 
5  d  and  5  d,  may  be  found.  Add  the  pressure  of  that  portion 
of  the  arch  which  is  included1  between  i  and  5  to  half  the 
weight  of  the  centre,  and  consider  this  amount  concentrated 
at  d,  and  acting  in  a  vertical  direction  ;  then,  by  constructing 
the  parallelogram  of  forces,  the  pressure  upon  dj  may  be  as- 
certained. 

414. — The  strains  having  been  obtained,  the  dimensions  of 
the  several  pieces  in  the  frames  bad  and  5  c  d,  may  be  found 
by  computation,  as  directed  in  the  case  of  roof  trusses,  from 
Arts.  375  to  380.  The  tie-beams  ld,ld,  if  made  of  sufficient 
size  to  resist  the  compressive  strain  acting  upon  them  from  the 
load  at  5,  will  be  more  than  large  enough  to  resist  the  tensile 
strain  upon  them  during  the  laying  of  the  first  part  of  the 
arch-stones  below  a  and  c. 

415. — In  the  construction  of  arches,  the  voussoirs,  or  arch- 
stones,  are  so  shaped  that  the  joints  between  them  are  perpen- 
dicular to  the  curve- of  the  arch,  or  to  its  tangent  at  the  point 
at  which  the  joint  intersects  the  curve.  In  a  circular  arch,  the 
joints  tend  toward  the  centre  of  the  circle :  in  an  elliptical 
arch,  the  joints  may  be  found  by  the  following  process  : 


Fig.  260. 


304  AMERICAN    HOTT9"K-CAEPENTER. 

416. — To  find  the  direction  of  the  joints  for  an  elliptical 
arch.  A  joint  being  wanted  at  #,  (Fig.  260,)  draw  lines  from 
that  point  to  the  foci,/  and//  bisect  the  angle,/a/,  with  the 
line,  a  I ;  then  a  "b  will  be  the  direction  of  the  joint, 


Fig.  ML 


41 7. — To  find  the  direction  of  the  joints  for  a  parabolic  arch 
A  joint  being  wanted  at  #,  (Fig.  261,)  draw  a  e,  at  right  angles 
to  the  axis,  eg ;  make  eg  equal  to  c e,  and  join  a  and  g ;  draw 
a  A,  at  right  angles  to  ag ;  then  a  h  will  be  the  direction  of 
the  joint.  The  direction  of  the  joint  from  5  is  found  in  -the 
same  manner.  The  lines,  a  g  and  bf,  are  tangents  to  the  curve 
at  those  points  respectively ;  and  any  number  of  joints  in  the 
curve  may  be  obtained,  by  first  ascertaining  the  tangents,  and 
then  drawing  lines  at  right  angles  to  them. 


Fig.  203. 

418. — Fig.  262  shows  a  simple  and  quite  strong  method  of 
lengthening  a  tie-beam ;  but  the  strength  consists  wholly  in 
the  bolts,  and  in  the  friction  of  the  parts  produced  by  screwing 
the  pieces  firmly  together.  Should  the  timber  shrink  to  even 
a  small  degree,  the  strength  would  depend  altogether  on  the 
bolts.  It  would  be  made  much  stronger  by  indenting  the 
pieces  together ;  as  at  the  upper  edge  of  the  tie-beam  in  Fig. 
263  ;  or  by  placing  keys  in  the  joints,  as  at  the  lower  edge  in 


FRAMING.  305 

the  same  figure.     This  process,  however,  weakens  the  beam  in 
proportion  to  the  depth  of  the  indents. 


Fig.  268. 


419. — Fig.  264  shows  a  method  of  scarfing,  or  splicing,  a 
tie-beam  without  bolts.    The  keys  are  to  be  of  well-seasoned, 


Fig.  264 


nard  wood,  and,  if  possible,  very  cross-grained.  The  addition 
of  bolts  would  make  this  a  very  strong  splice,  or  even  white- 
oak  pins  would  add  materially  to  its  strength. 


Fig.  26B. 

420. — Fig.  265  shows  about  as  strong  a  splice,  perhaps,  as 
can  well  be  made.  It  is  to  be  recommended  for  its  simplicity ; 
as,  on  account  of  there  being  no  oblique  joints  in  it,  it  can  be 
readily  and  accurately  executed.  A  complicated  joint  is  the 
worst  that  can  be  adopted ;  still,  some  have  proposed  joints 
that  seem  to  have  little  else  besides  complication  to  recom 
mend  them. 

421. — In  proportioning  the  parts  of  these  scarfs, 'the  depths 
of  all  the  indents  taken  together  should  be  equal  to  one-third 
of  the  depth  of  the  beam.  In  oak,  ash  or  elm,  the  whole 
length  of  the  scarf  should  be  six  times  the  depth,  or  thickness, 
of  the  beam,  when  there  are  no  bolts ;  but,  if  bolts  instead  of 
indents  are  used,  then  three  times  the  breadth  ;  and,  when  both 
methods  are  combined,  twice  the  depth  of  the  beam.  The 


306  AMERICAN   HOUSE-CARPENTER. 

length  of  the  scarf  in  pine  and  similar  soft  woods,  depending 
wholly  on  indents,  should  be  about  12  times  the  thickness,  or 
depth,  of  the  beam ;  when  depending  wholly  on  bolts,  6  times 
the  breadth ;  and,  when  both  methods  are  combined,  4  times 
the  depth. 


Fig.  266. 

422. — Sometimes  beams  have  to  be  pieced  that  are  required 
to  resist  cross  strains — such  as  a  girder,  or  the  tie-beam  of  a 
roof  when  supporting  the  ceiling.  In  such  beams,  the  fibres 
of  the  wood  in  the  upper  part  are  compressed ;  and  therefore 
a  simple  butt  joint  at  that  place,  (as  in  Fig.  266,)  is  far  prefer- 
able to  any  other.  In  such  case,  an  oblique  joint  is  tlie  very 
worst.  The  under  side  of  the  beam  being  in  a  state  of  tension, 
it  must  be  indented  or  bolted,  or  both ;  and  an  iron  plate  un- 
der the  heads  of  the  bolts,  gives  a  great  addition  of  strength. 

Scarfing  requires  accuracy  and  care,  as  all  the  indents  should 
bear  equally  ;  otherwise,  one  being  strained  more  than  another, 
there  would  be  a  tendency  to  splinter  off  the  parts.  Hence 
the  simplest  form  that  will  attain  the  object,  is  by  far  the  best. 
In  all  beams  that  are  compressed  endwise,  abutting  joints, 
formed  at  right  angles  to  the  direction  of  their  length,  are  at 
once  the  simplest  and  the  best.  For  a  temporary  purpose,  Fig. 
262  would  do  very  well ;  it  would  be  improved,  however,  by 
having  a  piece  bolted  on  all  four  sides.  Fig.  263,  and  indeed 
each  of  the  others,  since  they  have  no  oblique  joints,  would 
resist  compression  well. 

423. — In  framing  one  beam  into  another  for  bearing  pui 
poses,  such  as  a  floor-beam  into  a  trimmer,  the  best  place  to 
make  the  mortice  in  the  trimmer  fe  in  the  neutral  line,  (Arts 
317,  318,)  which  is  in  the  middle  of  its  depth.  Some  havo 
thought  that,  as  the  fibres  of  the  upper  edge  are  compressed,  a 


FRAMING.  307 

mortice  might  be  made  there,  and  the  tenon  driven  in  tight 
enough  to  make  the  parts  as  capable  of  resisting  the  compres- 
sion, as  they  would  be  without  it ;  and  they  have  therefore 
concluded  that  plan  to  be  the  best.  This  could  not  be  the  case, 
even  if  the  tenon  would  not  shrink ;  for  a  joint  between  two 
pieces  cannot  possibly  be  made  to  resist  compression,  so  well 
as  a  solid  piece  without  joints.  The  proper  place,  therefore, 
for  the  mortice,  is  at  the  middle  of  the  depth  of  the  beam ;  but 
the  best  place  for  the  tenon,  in  the  floor-beam,  is  at  its  bottom 
edge.  For  the  nearer  this  is  placed  to  the  upper  edge,  the 
greater  is  the  liability  for  it  to  splinter  oif ;  if  the  joint  is 


Fig.2«l. 

formed,  therefore,  as  at  Fig.  267,  it  will  combine  all  the  ad- 
vantages that  can  be  obtained.  Double  tenons  are  objection- 
able, because  the  piece  framed  into  is  needlessly  weakened, 
and  the  tenons  are  seldom  so  accurately  made  as  to  bear 
equally.  For  this  reason,  unless  the  tusk  at  a  in  the  figure  fits 
exactly,  so  as  to  bear  equally  with  the  tenon,  it  had  better  be 
omitted.  And  in  sawing  the  shoulders,  care  should  be  taken 
not  to  saw  into  the  tenon  in  the  least,  as  it  would  wound  the 
beam  in  the  place  least  able  to  bear  it. 

424:. — Thus  it  will  be  seen  that  framing  weakens  both  pieces, 
more  or  less.  It  should,  therefore,  be  avoided  as  much  as  pos- 
sible ;  and  where  it  is  practicable  one  piece  should  rest  upon 
the  other,  rather  than  be  framed  into  it.  This  remark  applies 
to  the  bearing  of  floor-beams  on  a  girder,  to  the  purlins  and 
jack-rafters  of  a  roof,  tfcc. 

425. — In  a  framed  truss  for  a  roof,  bridge,  partition,  &c., 
the  joints  should  be  so  constructed  as  to  direct  the  pressures 


308 


AMERICAN    HOUSE-CARPENTER. 


through  the  axes  of  the  several  pieces,  and  also  to  avoid  every 
tendency  of  the  parts  to  slide.    To  attain  this  object,  the  abut- 


Fig.  270. 


ting  surface  on  the  end  of  a  strut  should  be  at  right  angles  to 
the  direction  of  the  pressure ;  as  at  the  joint  shown  in  Fig. 
268  for  the  foot  of  a  rafter,  (see  Art.  277,)  in  Fig.  269  for  the 
head  of  a  rafter,  and  in  Fig.  270  for  the  foot  of  a  strut  or 
brace.  The  joint  at  Fig.  268  is  not  cut  completely  across  the 
tie-beam,  but  a  narrow  lip  is  left  standing  in  the  middle,  and 
a  corresponding  indent  is  made  in  the  rafter,  to  prevent  the 
parts  from  separating  sideways.  The  abutting  surface  should 
be  made  as  large  as  the  attainment  of  other  necessary  objects 
will  admit.  The  iron  strap  is  added  to  prevent  the  rafter  slid- 
ing out,  should  the  end  of  the  tie-beam,  by  decay  or  otherwise, 
splinter  oif.  In-  making  the  joint  shown  at  Fig.  269,  it  should 
be  left  a  little  open  at  a,  so  as  to  bring  the  parts  to  a  fair  bear- 
ing at  the  settling  of  the  truss,  which  must  necessarily  take 
place  from  the  shrinking  of  the  king-post  and  other  parts.  If 
the  joint  is  made  fair  at  first,  when  the  truss  settles  it  will  cause 
it  to  open  at  the  under  side  of  the  rafter,  thus  throwing  the 
whole .  pressure  upon  the  sharp  edge  at  a.  This  will  cause  an 
indentation  in  the  king-post,  by  which  the  truss  will  be  made 
to  settle  further ;  and.  this  pressure  not  being  in  the  axis  of  the 
rafter,  it  will  be  greatly  increased,  thereby  rendering  the  rafter 
liable  to  split  and  break. 

426. — If  the  rafters  and  struts  were  made  to  abut  end  to 
end,  as  in  Figs.  271,  272  and  273,  and  the  king  or  queen  post 
notched  on  in  halves  and  bolted,  the  ill  effects  of  shrinking 


309 


would  be  avoided.     This  method  has  been  practised  with  suc- 
cess, in  some  of  the  most  celebrated  bridges  and  roofs  in  Eu- 


Fig.  2TL 


Fig.  2T2. 


Fig.  2T8. 


rope  ;  and,  were  its  use  adopted  in  this  country,  the  unseemly 
sight  of  a  hogged  ridge  would  seldom  be  met  with.  A  plate 
of  cast  iron  between  the  abutting  surfaces  will  equalize*  the 
pressure. 


Fig.  2T4 


Fig.  275. 


427. — Fig.  274  is  a  proper  joint  for  a  collar-beam  in  a  small 
roof:  the  principle  shown  here  should  characterize  all  tie- 
joints.  The  dovetail'  joint,  although  extensively  practised  in 
the  above  and  similar  cases,  is  the  very  worst  that  can  be  em- 
ployed. The  shrinking  of  the  timber,  if  only  to  a  small  de- 
gree, permits  the  tie  to  withdraw — as  is  shown  at  Fig.  275. 
The  dotted  line  shows  the  position  of  the  tie  after  it  haa 
shrunk. 

428. — Locust  and  white-oak  pins  are  great  additions  to  the 
strength  of  a  joint.  In  many  cases  they  would  supply  the 
place  of  iron  bolts ;  and,  on  account  of  their  small  cost,  they 
should  be  used  in  preference  wherever  the  strength  of  iron  is 


310  AMERICAN   HOUSE-CAKPENTER. 

not  requisite.  In  small  framing,  good  cut  nails  are  of  great 
service  at  the  joints ;  but  they  should  not  be  trusted  to  bear 
any  considerable  pressure,  as  they  are  apt  to  be  brittle.  Iron 
straps  are  seldom  necessary,  as  all  the  joinings  in  carpentry 
may  be  made  without  them.  They  can  be  used  to  advantage, 
however,  at  the  foot  of  suspending-pieces,  and  for  the  rafter  at 
the  end  of  the  tie-beam.  In  roofs  for  ordinary  purposes,  the 
iron  straps  for  suspending-pieces  may  be  as  follows :  When  the 
longest  unsupported  part  of  the  tie-beam  is 

10  feet,  -the  strap  may  be  1  inch  wide  by  •&  thick. 
15     "          "  "      1£  "  "       £     " 

20    "          "  "     2     "  "       i     " 

In  fastening  a  strap,  its  hold  on  the  suspending-piece  will  be 
much  increased  by  turning  its  ends  into  the  wood.  Iron  straps 
should  be  protected  from  rust ;  for  thin  plates  of  iron  decay 
very  soon,  especially  when  exposed  to  dampness.  For  this 
purpose,  as  soon  as  the  strap  is  made,  let  it  be  heated  to  about 
a  blue  heat,  and,  while  it  is  hot,  pour  over  its  entire  surface 
raw  linseed  oil,  or  rub  it  with  beeswax.  Either  of  these'will 
give  it  a  coating  which  dampness  will  not  penetrate. 


EBON   GIBBERS. 


1 


Fig.  27«L  Fig.  277. 

429.—^.  276  represents  the  front  view,  and  Fig.  277  the 
cross  section  at  middle,  of  a  cast  iron  girder  of  proper  form 
for  sustaining  a  weight  equally  diffused  over  its  length.  The 
curve  is  that  of  a  parabola :  generally  an  arc  of  a  circle  ia 


FRAMING.  311 

used,  and  is  near  enough.  Beams  of  this  form  are  much  used 
to  sustain  brick  walls  of  buildings ;  the  brickwork  resting  upon 
the  bottom  flange,  and  laid,  not  arching,  but  horizontal.  In 
the  cross  section,  the  bottom  flange  is  made  to  contain  in  area 
four  times  as  much  as  the  top  flange.  The  strength  will  be  in 
proportion  to  the  area  of  the  bottom  flange,  and  to  the  height 
or  depth.  Hence,  to  obtain  the  greatest  strength  from  a  given 
amount  of  material,  it  is  requisite  to  make  the  upright  part,  or 
the  blade,  rather  thin  ;  yet,  in  order  to  prevent  injurious  strains 
in  .the  casting  while  it  is  cooling,  the  parts  should  be  nearly 
equal  in  thickness.  The  thickness  of  the  three  parts — blade, 
top  flange  and  bottom  flange,  may  be  made  in  proportion  as  5, 
6  and  8.  For  a  beam  of  this  form,  the  weight  equally  dif- 
fused over  it  equals 

^,  =  9000^.  (199.) 

The  depth  equals 

d  =      lw    .  (200.) 

9000  t  a 

The  area  of  the  bottom  flange  equals 

lw  /C.M  ^ 

a  =  906073-  (201° 

where  w  equals  the  weight  in  pounds  equally  diffused  over  the 
length ;  d,  the  depth,  or  height  in  inches  of  the  cross  section 
at  middle ;  «,  the  area  of  the  bottom  flange  in  inches ;  /,  the 
length  of  the  beam  in  feet,  in  the  clear  between  the  bearings ; 
and  £,  a  decimal  in  proportion  to  unity  as  the  safe  weight  is  to 
the  breaking  weight.  This  is  usually  from  0'2  to  0'3,  or  from 
one-fifth  to  one-third,  at  discretion. 

430. — Beams  of  this  form,  laid  in  series,  are  much  used  in 
sustaining  brick  arches  turned  over  vaults  and  other  fire-proof 
rooms,  forming  a  roof  to  the  vault  or  room,  and  a  floor  above ; 
the  arches  springing  from  the  flanges,  one  on  either  side  of  the 
beam,  as  in  Fig.  278. 


312 


AMERICAN   HOUSE-CARPENTER. 


For  this  use  the  depth  of  cross  section  at  middle  equals 

(202.) 


(203.) 


9000  t  a 
The  area  of  the  bottom  flange  equals 

of  I* 

9000  t  $ 

where  the  symbols  signify  as  before,  and  c  equals  the  distance 
apart  from  centres  in  feet  at  which  the  beams  are  placed,  and 
/  the  weight  per  superficial  foot,  in  pounds,  including  the 
weight  of  the  material  of  which  the  floor  is  constructed. 


Practical  Rules  and  Examples. 

431. — For  a  single  girder  the  dimensions  may  be  found  by 
the  following  rule,  ((200)  and  (201) :) 

Rule  LXYIII. — Divide  the  weight  in  pounds  equally  dif- 
fused over  the  length  of  the  girder  by  a  decimal  in  proportion 
to  unity  as  the  safe  weight  is  to  the  breaking  weight,  multiply 
the  quotient  by  the  length  in  feet,  and  divide  the  product  by 
9000.  Then  this  quotient,  divided  by  the  depth  of  the  beam 
at  middle,  will  give  the  area  of  the  bottom  flange ;  or,  if  di- 
vided by  the  area  of  the  bottom  flange,  will  give  the  depth—- 
the area  and  depth  both  in  inches. 

Example. — Let  the  weight  equally  diffused  over  a  girder 
equal  60000  pounds ;  the  decimal  that  is  in  proportion  to  unitv 
as  the  safe  weight  is  to  be  to  the  breaking  weight,  equal  0*3 


FRAMING.  313 

the  length  in  the  clear  of  the  bearings  equal  20  feet.  Then 
60000  divided  by  0-3  equals  200000,  and  this  by  20  ( equals 
4000000 ;  this  divided  by  9000  equals  444£.  Now  if  the*  depth, 
is  fixed,  say  at  20  inches,  then  444$,  divided  by  20,  equals  22|, 
equals  the  area  of  the  bottom  flange  in  inches.  But  if  the 
area  is  given,  say  24  inches,  then  to  find  the  depth,  divide 
444£  by  24,  and  the  quotient,  18*5,  equals  the  depth  in  inches ; 
and  such  a  girder  may  be  made  with  a  bottom  flange  of  2  by 
12  inches,  top  flange,  (equal  to  \  of  bottom  flange,)  1^  by  4 
inches,  and  the  blade  1$  inches  thick. 

432. — For  a  series  of  girders  or  iron  beams,  the  dimensions 
may  be  found  by  the  following  rule :  (202)  and  (203). 

Rule  LXIX. — Divide  the  weight  per  superficial  foot,  in 
pounds,  by  a  decimal  in  proportion  to  unity  as  the  safe  weight 
is  to  the  breaking  weight,  and  multiply  the  quotient  by  the 
square  of  the  length  of  the  beams  and  by  the  distance  apart  at 
which  the  beams  are  placed  from  centres,  both  in  feet,  and  di- 
vide the  product  by  9000.  Then  this  quotient,  divided  by  the 
depth  of  the  beams  at  middle,  will  give  the  area  of  the  bottom 
flange ;  or,  if  divided  by  the  area  of  the  bottom  flange,  will 
give  the  depth  of  the  beam — the  depth  and  area  both  in  inches. 

Example. — Let  the  weight  per  superficial  foot  resting  upon 
an  arched  floor  be  200  pounds,  and  the  weight  of  the  arches, 
concrete,  &c.,  equal  100  pounds,  total  300  pounds  per  superfi- 
cial foot.  Let  the  proportion  of  the  breaking  weight  to  be 
trusted  on  the  beams  equal  0*3,  the  length  of  the  beams  in  the 
clear  of  the  bearings  equal  12  feet,  and  the  distance  apart  from 
centres  at  which  they  are  placed  equal  4  feet.  Then  300  di- 
vided by  0-3  equals  1000 ;  this  multiplied  by  144  (the  square 
of  12),  equals  144000,  and  this  by  4,  equals  576000  ;  this  di- 
vided by  9000,  equals  64.  Now  if  the  depth  is  fixed,  and  at 
8  inches,  then  64  divided  by  8  equals  8,  equals  the  area  of  the 
bottom  flange.  But  if  the  area  of  the  bottom  flange  is  fixed, 
and  at  6  inches,  then  64,  divided  by  6,  equals  10f,  the  depth 
40 


314 


AMERICAN   HOUSE-CARPENTEK. 


required.  Such  a  beam  may  be  made  with  the  bottom  flange 
1  by  6  inches,  the  top  flange,  (equal  to  one-quarter  of  the  bot- 
tom flange,)  £  by  2  inches,  and  the  blade  £  inch  thick. 

433. — The  kind  of  girder  shown  at  Fig.  280,  (a  cast  iron 
arch  with  a  wrought  iron  tie  rod,)  is  extensively  used  as  a  sup- 
port upon  which  to  build  brick  walls  where  the  space  below  is 
required  to  be  free.  The  objections  to  its  use  are,  the  dispro- 
portion between  the  material  and  the  strains,  and  the  enhanced 
cost  over  the  cast  iron  girder  formed  as  in  Figs.  276  and  277. 
The  material  in  the  cast  arch,  (Fig.  280,)  is  greatly  in  excess 
over  the  amount  needed  to  resist  effectually  the  compressive 
strains  induced  by  the  load  through  the  axis  of  the  arch,  while 
the  wrought  metal  in  the  tie  is  usually  much  less  than  is  re- 
quired to  resist  the  horizontal  thrust  of  the  arch ;  absolute  fail- 
ure being  prevented,  partly  by  the  weight  of  the  walls  resting 
on  the  haunches,  and  partly  by  the  presence  of  adjoining 
buildings,  their  walls  acting  as  buttresses  to  the  arch.  Some 
instances  have  occurred  where  the  tie  has  parted. 

Where  this  arched  girder  is  used  it  is  customary  to  lay  the 
first  courses  of  brick  in  the  form  of  an  arch.  This  brick  arch 
of  itself  is  quite  sufficient  to  sustain  the  compressive  strain, 
and,  were  there  proper  resistance  to  the  horizontal  thrust  pro- 
vided, the  brick  arch  would  entirely  supersede  the  necessity 
for  the  girder.  Indeed,  the  instances  are  not  rare  where  con- 
structions of  this  nature  have  proved  quite  satisfactory,  the 
horizontal  thrust  of  the  arch  being  sustained  by  a  tie  rod 
secured  to  a  pair  of  cast  iron  heel  plates,  as  in  Fig.  279.  The 


Fig.  279. 


FRAMING.  315 

brick  arch,  in  this  case,  being  built  upon  a  wooden  centre, 
which  was  afterwards  removed. 

The  diameter  of  the  rod  required  for  an  arch  of  this  kind  is 
equal  to 


where  w  equals  the  weight  in  pounds  equally  diffused  over  the 
arch  ;  5,  the  length  of  the  rod,  clear  of  the  heel  plates,  in  feet ; 
and  A,  the  height  at  the  middle,  or  rise,  of  the  arc,  in  inches ; 
X>,  the  diameter,  being  also  in  inches. 

"When  the  diameter  found  by  formula  (204)  is  impracticably 
large,  this  difficulty  may  be  overcome  by  dividing  the  metal 
into  two  rods.  In  the  bow-string  girder,  (Fig.  280,)  two  rods 
cannot  be  used  with  advantage,  because  of  the  difficulty  in 
adjusting  their  lengths  so  as  to  ensure  to  each  an  equal  amount 
of  the  strain.  But  in  the  case  of  the  brick  arch,  the  two  heel 
plates  being  disconnected,  any  discrepancy  of  length  in  the- 
rods  is  adjusted  simply  by  the  pressure  of  the  arch  acting  on 
the  plates.  When  there  are  to  be  two  rods,  the  diameter  of 
each  rod  equals 

(205.) 


Practical  Rule  and  Example. 

434. — To  obtain  the  diameter  of  wrought  iron  tie-rods  foi 
heel  plates,  as  in  Fig.  279,  proceed  by  this  rule. 

Rule  LXX. — Multiply  the  weight  in  pounds  equally  distri- 
buted over  the  arch  by  the  length  of  the  tie-rod  in  feet,  clear 
of  the  heel  plates,  and  divide  the  product  by  the  height  of  the 
arc  in  inches,  (that  is,  the  height  at  the  middle,  from  the  axia 
of  the  tie-rod  to  the  centre  of  the  depth  of  the  brick  arch,)  then, 
if  there  is  to  be  but  one  tie- rod,  divide  the  quotient  by  3000 ; 


316 


AMEEICAN    HOUSK-CABPENTER. 


but  if  two,  then  divide  by  6000,  and  the  square  root  of  the 
quotient,  in  either  case,  will  be  the  required  diameter. 

Example. — The  weight  to  be  supported  on  a  brick  arch, 
equally  distributed,  is  24000  pounds ;  the  length  of  the  tie-rod, 
clear  of  the  heel  plates,  is  10  feet ;  and  the  height,  or  rise,  of 
the  arc  is  10  inches.  Now  by  the  rule,  24000  X  10  =  240000. 
This  divided  by  10,  equals  24000.  Upon  the  presumption  that 
one  tie-rod  only  will  be  needed,  divide  by  3000,  and  the  quo- 
tient is  8,  the  square  root  of  which  is  2-82  inches.  This  is 
rather  large,  therefore  there  had  better  be  two  rods.  In  this 
case  the  quotient,  24000,  divided  by  6000,  equals  4,  the  square 
root  of  which  is  2,  the  diameter  required.  The  arch  should, 
therefore,  have  two  rods  of  2  inches  diameter.  Two  rods  are 
preferable  to  one.  The  iron  is  stronger  per  inch  in  small  rods 
than  in  large  ones,  and  the  rules  require  no  more  metal  in  the 
two  rods  than  in  the  one. 


Fig.  280. 

435. — The  Bow-string  Girder,  as  per  Fig.  280,  has  little  to 
recommend  it,  (see  Art.  433,)  yet  because  it  has  by  some  been 
much  used,  it  is  well  to  show  the  rules  that  govern  its  strength, 
if  only  for  the  benefit  of  those  who  are  willing  to  be  governed 
by  reason  rather  than  precedent.  To  resist  the  horizontal 
thrust  of  the  cast  arch,  the  diameter  of  the  rod  must  equal 
(204) 


But  the  cast  iron  arch  has  a  certain  amount  of  strength  to  re 


FRAMING.  317 

sist  cross  strains :  this  strength  must  be  considered.  Upon  the 
presumption  that  the  cross  section  of  the  cast  arch  at  the  mid- 
dle is  of  the  most  favorable  form,  as  in  Fig.  277,  or  at  least 
that  it  have  a  bottom  flange,  (although  the  most  of  those  cast 
are  without  it),  the  strength  of  the  cast  arch  to  resist  cross 
strains  is  shown  by  formula  (199),  when  Z,  its  length,  is  changed 
to  5,  its  span.  The  weight  in  pounds  equally  diffused  over  the 
arch  will  then  equaj 

_  9000  t  a  d 

8 

This  is  the  weight  borne  by  the  cast  arch  acting  simply  as  a 
beam.  Deducting  this  weight  from  the  whole  weight,  the  re- 
mainder is  the  weight  to  be  sustained  by  the  rod.  Calling  the 
whole  weight  w,  then 

9000  tad     ws-  9000  tad  =  W 

w  — =  — 

s  s 

Therefore,  from  (204),  the  diameter* equals 


=v 


3000  A 
w  s  —  9000  t  a  d 


(206.) 


3000  A 

where  D  equals  the  diameter  of  the  rod  in  inches ;  w,  the 
weight  in  pounds  equally  diffused  over  the  arch ;  s,  the  span 
of  the  arch  in  feet ;  A,  the  rise  or  height  of  the  arc  at  middle, 
in  inches ;  <?,  the  height  or  depth  of  the  cross  section  of  the 
cast  arch  in  inches ;  «,  the  area  of  the  bottom  flange  of  the 
cross  section  of  the  cast  arch  in  inches ;  and  t,  a  decimal  in 
proportion  to  unity  as  the  safe  weight  is  to  be  to  the  breaking 
weight. 

The  rule  in  words  at  length,  is 

RuU  LXXL — Multiply  the  decimal  in  proportion  to  unity 


318  AMERICAN   HOUSE-CARPENTER. 

as  the  safe  weight  is  to  be  to  the  breaking  weight,  by  f»00(? 
times  the  depth  of  the  cross  section  of  the  cast  arch  at  middle, 
and  by  the  area  of  the  bottom  flange  of  said  section,  both  in 
inches,  and  deduct  the  product  from  the  weight  in  pounds 
equally  diffused  over  the  arch  multiplied  by  the  span  in  feet, 
and  divide  the  remainder  by  3000  times  the  height  of  the  arc 
in  inches,  measured  from  the  axis  of  the  tie-rod  to  the  centre 
of  the  depth  of.  the  cast  arch  at  middle,  and  the  square  root 
of  the  quotient  will  be  the  diameter  of  the  rod  in  inches. 

Example. — The  rear  wall  of  a  building  is  of  brick,  and  is  40 
feet  high,  and  21  feet  wide  in  the  clear  between  the  piers  of 
the  story  below.  Allowing  for  the  voids  for  windows,  this 
wall  will  weigh  about  63000  pounds ;  and  it  is  proposed  to 
support  it  by  a  bow-string  girder,  of  which  the  cross  section  at 
middle  of  the  cast  arch  is  8  inches  deep,  and  has  a  bottom 
flange  containing  12  inches  area.  The  rise  of  the  curve  or  arc 
is  24  inches.  What  must  be  the  diameter  of  the  rod,  the  por- 
tion of  the  breaking  weight  of  the  cast  arch,  considered  safe 
to  trust,  being  three-tenths  or  0'3  ?  By  the  rule,  0*3  x  9000 
X  8  x  12  =  259200 ;  then  63000  x  21  -  259200  =  10638.00. 
This  remainder  divided  by  (3000  x  24  =)  72000,  the  quotient 
equals  14*775  ;  the  square  root  of  which,  3'84,  or  nearly  3| 
inches,  is  the  required  diameter. 

This  size,  though  impracticably  large,  is  as  small  as  a  due 
regard  for  safety  will  permit ;  yet  it  is  not  unusual  to  find  the 
rods  in  girders  intended  for  as  heavy  a  load  as  in  this  exam- 
ple, only  2^  and  2£  inches !  Were  it  possible  to  attach  the 
rod  so  as  not  to  injure  its  strength  in  the  process  of  shrinking 
it  in — putting  it  to  its  place  hot,  and  depending  on  the  con- 
traction of  the  metal  in  cooling  to  bring  it  to  a  proper  bearing 
— and  were  it  possible  to  have  the  bearings  so  true  as  to  induce 
the  strain  through  the  axis  of  the  rod,  and  not  along  its  side, 
(Art.  308,)  then  a,  less  diameter  than  that  given  by  the  rule 
would  suffice.  But  while  these  contingencies  remain,  the  rule 


FRAMING.  319 

cannot  safely  be  reduced,  for,  in  the  rule,  the  value  of  jT,  for 
wrought  iron,  (Table  III.,  Art.  308,)  is  taken  at  nearly  600C 
pounds,  a  point  rather  high  in  consideration  of  the  size  of  the 
rod  and  the  injuries,  before  stated,  to  which  it  is  subjected. 
In  cases  where  a  girder  wholly  of  cast  iron  (Fig.  276)  is  not 
preferred,  it  were  better  to  build  a  brick  arch  resting  on  heel 
plates,  (Fig.  279,)  in  which  the  metal  required  to  resist  the 
thrust  may  be  divided  into  two  rods  instead  of  oi\e,  thus  render- 
ing the  size  more  practical,  and  at  the  same  time  avoiding  the 
injuries  to  which  rods  in  arch  girders  are  subjected.  The  heel- 
plate arch  is  also  to  be  preferred  to  the  cast  arch  on  the  score 
of  economy ;  inasmuch  as  the  brick  which  is  substituted  for 
the  east  arch  will  cost  less  than  iron.  For  example,  suppose 
the  cross  section  of  the  iron  arch  to  be  thus :  the  blade  or  up- 
right part  8  by  1£  inches,  the  top  flange  12  by  1J  inches,  and 
the  bottom  flange  6  by  If  inches.  At  these  dimensions,  the 
a^ea  of  the  cross  section  will  equal  12  +  15  +  10£  =  37^ 
inches.  A  bar  of  cast  iron,  one  foot  long  and  one  inch  square, 
will  weigh  3'2  pounds;  therefore,  37£  X  3*2  =  120  pounds, 
equals  the  weight  of  the  cast  arch  per  lineal  foot  The  price 
of  castings  per  pound,  as  also  the  price  of  brickwork  per. 
cubic  foot,  of  course  will  depend  upon  the  locality  and  the 
state  of  the  market  at  the  time,  but  for  a  comparison  they  may 
be  stated,  the  one  at  three  and  a  half  cents  -per  pound,  and  the 
other  at  thirty  cents  per  cubic  foot.  At  these  prices  the  cast 
arch  will  cost  120  x  3£  =  $4  20  per  lineal  foot;  while  the 
brick  arch — 12  inches  high  and  12  inches  thick — will  cost  30 
^cents  per  lineal  foot.  The  difference  is  $3  90.  This  amount 
is  not  all  to  be  credited  to  the  account  of  the  brick  arch, 
Proper  allowance  is  to  be  made  for  the  cost  of  the  heel  plates, 
and  of  the  wooden  centre ;  also  for  the  cost  of  a  small  addi- 
tion to  the  size  of  the  tie  rods,  which  is  required  to  sustain  the 
strain  otherwise  borne  by  the  cast  arch  in  its  resistance  to  a 
cross  strain  (Art.  435).  Deducting  the  cost  of  these  items, 


AMERICAN   HOUSE-CARPENTER. 


the  difference  in  favor  of  the  brick  arch  will  be  about  $3  pel 
foot.  This,  on  a  girder  25  feet  long,  amounts  to  $75.  The 
difference  in  all  cases  will  not  equal  this,  but  will  be  sufficiently 
great  to  be  worth  saving. 


SECTION  V — DOORS,  WINDOWS,  dtc. 


DOORS. 

436. — Among  the  several  architectural  arrangements  of  an  edi- 
fice, the  door  is  by  no  means  the  least  in  importance  ;  and,  if  pro- 
perly constructed,  it  is  not  only  an  article  of  use,  but  also  of  or- 
nament, adding  materially  to  the  regularity  and  elegance  of  the 
apartments.  The  dimensions  and  style  of  finish  of  a  door,  should 
be  in  accordance  with  the  size  and  style  of  the  building,  or  the 
apartment  for  which  it  is  designed.  As  regards  the  utility  of 
doors,  the  principal  door  to  a  public  building  should  be  of  suffi- 
cient width  to  admit  of  a  free  passage  for  a  crowd  of  people ; 
while  that  of  a  private  apartment  will  be  wide  enough,  if  it  per- 
mit one  person  to  pass  without  being  incommoded.  Experience 
has  determined  that  the  least  width  allowable  for  this  is  2  feet  8 
inches  ;  although  doors  leading  to  inferior  and  unimportant  rooms 
may,  if  circumstances  require  it,  be  as  narrow  as  2  feet  6  inches ; 
and  doors  for  closets,  where  an  entrance  is  seldom  required,  may 
be  but  2  feet  wide.  The  width  of  the  principal  door  to  a  public 
building  may  be  from  6  to  12  feet,  according  to  the  size  of  the 
building  ;  and  the  width  of  doors  for  a  dwelling  may  be  from  2 
feet  8  inches,  to  3  feet  6  inches.  If  the  importance  of  an  apart- 
ment in  a  dwelling  be  such  as  to  require  a  door  of  greater  width 


322  AMERICAN7    HOUSE-CAIIPENTER. 

than  3  feet  6  inches,  the  opening  should  be  closed  with  two 
doors,  or  a  door  in  two  folds ;  generally,  in  such  cases,  where  the 
opening  is  from  5  to  8  feet,  folding  or  sliding  doors  are  adopted, 
As  to  the  height  of  a  door,  it  should  in  no  case  be  less  than  about 
0  feet  3  inches ;  and  generally  not  less  than  6  feet  8  inches. 

437. — The  proportion  between  the  width  and  height  of  single 
doors,  for  a  dwelling,  should  be  as  2  is  to  5 ;  and,  for  entrance- 
doors  to  public  buildings,  as  1  is  to  2.  If  the  width  is  given  and 
the  height  required  of  a  door  for  a  dwelling,  multiply  the  width 
by  5,  and  divide  the  product  by  2 ;  but,  if  the  height  is  given  and 
the  width  required,  divide  by  5,  and  multiply  by  2.  Where  two 
or  more  doors  of  different  widths  show  in  the  same  room,  it  is 
well  to  proportion  the  dimensions  of  the  more  important  by  the 
above  rule,  and  make  the  narrower  doors  of  the  same  height  as 
the  wider  ones ;  as  all  the  doors  in  a  suit  of  apartments,  except 
the  folding  or  sliding  doors,  have  the  best  appearance  when  of 
one  height.  The  proportions  for  folding  or  sliding  doors  should 
be  such  that  the  width  may  be  equal  to  |  of  the  height ;  yet  this 
rule  needs  some  qualification :  for,  if  the  width  of  the  opening 
be  greater  than  one-half  the  width  of  the  room,  there  will  not  be 
a  sufficient  space  left  for  opening  the  doors ;  also,  the  height 
should  be  about  one-tenth  greater  than  that  of  the  adjacent  single 
doors. 

438. — Where  doors  have  but  two  panels  in  width,  let  the  stiles 
and  muntins  be  each  4  °f  the  width  ;  or,  whatever  number  of 
panels  there  may  be,  let  the  united  widths  of  the  stiles  and  the 
muntins,  or  the  whole  width  of  the  solid,  be  equal  to  |  of  the  width 
of  the  door.  Thus  :  in  a  door,  35  inches  wide,  containing  two 
panels  in  width,  the  stiles  should  be  5  inches  wide  ;  and  in  a  door, 
3  feet  6  inches  wide,  the  stiles  should  be  6  inches.  If  a  door,  3 
feet  6  inches  wide,  is  to  have  3  panels  in  width,  the  stiles  and 
muntins  should  be  each  4£  inches  wide,  each  panel  being  8  inches. 
The  bottom  rail  and  the  lock  rail  ought  to  be  each  equal  in 
*ridth  to  TV  of  the  height,  of  the  door ;  and  the  top  rail,  and  all 


DOORS,    WINDOWS,    &C. 


others,  of  the  same  width  as  the  stiles.     The  moulding  on  the 
panel  should  be  equal  in  width  to  ?  of  the  width  of  the  stiltv 


Fig.  281. 

439. — pig,  281  shows  an  approved  method  of  trimming  doors : 
a  is  the  door  stud  ;  6,  the  lath  and  plaster  ;  c,  the  ground  ;  d,  the 
jamb ;  e,  the  stop  ;  f  and  g,  architrave  casings  ;  and  7i,  the  door 
stile.  It  is  customary,  in  ordinary  work  to  form  the  stop  for  the 
door  by  rebating-  the  jamb.  But,  when  the  door  is  thick  and 
heavy,  a  better  plan  is  to  nail  on  a  piece  as  at  e  in  the  figure. 
This  piece  can  be  fitted  to  the  door,  and  put  on  after  the  door  is 
hung  ;  so,  should  the  door  be  a  trifle  winding,  this  will  correct 
the  evil,  and  the  door  be  made  to  shut  solid. 

440. — Fig.  282  is  an  elevation  of  a  door  and  trimmings  suita- 
ble for  the  best  rooms  of  a  dwelling.  (For  trimmings  generally, 
see  Sect.  III.)  The  number  of  panels  into  which  a  door  should 
be  divided,  is  adjusted  at  pleasure ;  yet  the  present  style  of  finish- 
ing requires,  that  the  number  be  as  small  as  a  proper  regard  for 
strength  will  admit.  In  some  of  our  best  dwellings,  doors  have 
been  made  having  only  two  upright  panels.  A  few  years  expe- 
lence,  however,  has  proved  that  the  omission  of  the  lock  rail 
is  at  the  expense  of  the  strength  and  durability  of  the  door ;  & 
four-panel  door,  therefore,  is  the  best  that  can  be  made. 

441. — The  doors  of  a  dwelling  should  all  be  hung  so  as  to  open 
into  the  principal  rooms  ;  and,  in  general,  no  door  should  be  hung 
to  open  into  the  hall,  or  passage.  As  to  the  proper  edge  of  the 
door  on  which  to  affix  the  hinges,  no  general  rule  can  be  assigned 


324 


AMERICAN   HOUSE-CARPENTER. 


Fig.  288. 


442. — A  window  should  be  of  such  dimensions,  and  in  such 
i  position,  as  to  admit  a  sufficienc3T  of  light  to  that  part  of  the 
ipartment  for  which  it  is  designed.  ~No  definite  rule  for  the  size 


DOORS,    WINDOWS,    &C.  325 

can  well  be  given,  that  will  answer  in  all  cases  ;  yet,  as  an  ap- 
proximation, the  following  has  been  used  for  general  purposes. 
Multiply  together  the  length  and  the  breadth  in  feet  of  the  apart- 
ment to  be  lighted,  and  the  product  by  the  height  in  feet ;  then 
the  square-root  of  this  product  will  show  the  required  number  of 
square  feet  of  glass. 

4:4:3. — To  ascertain  the  dimensions  of  window  frames,  add  4£ 
inches  to  the  width  of  the  glass  for  their  width,  and  6^-  inches  to 
the  height  of  the  glass  for  their  height.  These  give  the  dimen- 
sions, in  the  clear,  of  ordinary. frames  for  12-light  windows ;  the 
height  being  taken  at  the  inside  edge  of  the  sill.  In  a  brick  wall, 
the  width  of  the  opening  is  8  inches  more  than  the  width  of  the 
glass — 4£  for  the  stiles  of  the  sash,  and  3£  for  hanging  stiles — 
and  the  height  between  the  stone  sill  and  lintel  is  about  10 i  inches 
more  than  the  height  of  the  glass,  it  being  varied  according  to  the% 
thickness  of  the  sill  of  the  frame. 

444. — In  hanging  inside  shutters  to  fold  into  boxes,  it  is  ne- 
cessary to  have  the  box  shutter  about  one  inch  wider  than  the 
flap,  in  order  that  the  flap  may  not  interfere  when  both  are  folded 
into  the  box.  The  usual  margin  shown  between  the  face  of  the 
shutter  when  folded  into  the  box  and  the  quirk  of  the  stop  bead, 
or  edge  of  the  casing,  is  half  an  inch ;  and,  in  the  usual  method 
of  letting  the  whole  of  the  thickness  of  the  butt  hinge  into  the 
edge  of  the  box  shutter,  it  is  necessary  to  make  allowance  for  the 
throw  of  the  hinge.  This  may,  in  general,  be  estimated  at  i  of 
an  inch  at  each  hinging  ;  which  being  added  to  the  margin,  the 
entire  width  of  the  shutters  will  be  1^  inches  more  than  the  width 
of  the  frame  in  the  clear.  Then,  to  ascertain  the  width  of  the 
box  shutter,  add  1£  inches  to  the  width  of  the  frame  in  the  clear, 
between  the  pulley  stiles  ;  divide  this  product  by  4,  and  add 
half  an  inch  to  the  quotient ;  and  the  last  product  will  be  the  re- 
quired width.  For  example,  suppose  the  window  to  have  3 
lights  in  width,  11  inches  each.  Then,  3  times  11  is  33,  and  4£ 
added  for  the  wood  of  the  sash,  gives  37$ 37:J-  and  U  is  39 


326  AMERICAN    HOUSE  CARPENTER. 

and  39.  divided  by  4,  gives  9g  ;  to  which  add  half  an  inch,  and 
the  result  will  be  10£  inches,  the  width  required  for  the  box  shutter. 
445. — In  disposing  and  proportioning  windows  for  the  walls  of 
a  building,  the  rules  of  architectural  taste  require  that  they  be  of 
different  heights  in  different  stories,  but  of  the  same  width.  The 
windows  of  the  upper  stories  should  all  range  perpendicularly 
over  those  of  the  first,  or  principal,  story ;  and  they  should  be 
disposed  so  as  to  exhibit  a  balance  of  parts  throughout  the  front 
of  the  building.  To  aid  in  this,  it  is  always  proper  to  pl;ice  the 
front  door  in  the  middle  of  the  front  of  the  building  ;  and,  where 
the  size  of  the  house  will  admit  of  it,  this  plan  should  be  adopted. 
(See  the  latter  part  of  Art.  224.)  The  proportion  that  the  height 
should  bear  to  the  width,  may  be,  in  accordance  with  general 
usage,  as  follows  : 

^          The  height  of  basement  windows,  1^  of  the  width. 
"         "  principal-story     "     2j  " 

"         "  second-story         "     If  " 

"        "          third-story  "1^  ' 

"         "  fourth-story          "     H  " 

"         "  attic-story  "     the  same  as  the  width. 

But,  in  determining  the  height  of  the  windows  for  the  several 
stories,  it  is  necessary  to  take  into  consideration  the  height  of  the 
story  in  which  the  window  is  to  be  placed.  For,  in  addition  to 
the  height  from  the  floor,  which  is  generally  required  to  be  from 
28  to  30  inches,  room  is  wanted  above  the  head  of  the  window 
for  the  window-trimming  and  the  cornice  of  the  room,  besides 
some  respectable  space  which  there  ought  to  be  between  these. 

446. — Doors  and  windows  are  usually  square-headed,  or  termi- 
nate in  a  horizontal  line  at  top.  These  require  no  special  direc- 
tions for  their  trimmings.  But  circular-headed  doors  and  win- 
dows are  more  difficult  of  execution,  and  require  some  attention. 
If  the  jambs  of  a  door  or  window  be  placed  at  right  angles  to  the 
face  of  the  wall,  the  edges  of  the  soffit,  or  surface  of  the  head, 
would  be  straight,  and  its  length  be  found  by  getting  the 


DOORS,  WINDOWS,    AC. 


327 


stretch-out  of  the  circle,  (An.  92;)  but,  when  the  jaubs  are 
placed  obliquely  to  the  face  of  the  wall,  occasioned  by  the  de- 
mand for  light  in  an  oblique  direction,  the  form  of  the  soffit 
will  be  obtained  by  the  following  article :  and,  when  the  face 
of  the  wall  is  circular,  as  in  the  succeeding  one. 


Fig.  283. 

447. — To  find  the  form  of  the  soffit  for  circular  window 
heads,  when  the  light  is  received  in  an  oblique  direction.  Let 
abed,  (Fig:  283,)  be  the  ground-plan  of  a  given  window,  ande/ 
a,  a  vertical  section  taken  at  right  angles  to  the  face  of  the  jambs. 
From  a,  through  e,  draw  ag,  at  right  angles  to  a  b  ;  obtain  the 
stretch-out  of  ef  a,  and  make  e  g  equal  to  it ;  divide  e  g  and  e 
f  a,  each  into  a  like  number  of  equal  parts,  and  drop  perpen- 
diculars from  the  points  of  division  in  each  ;  from  the  points  of 
intersection,  1,  2,  3,  &c.,  in  the  line,  a  d,  draw  horizontal  lines  to 
meet  corresponding  perpendiculars  from  eg;  then  those  points 
of  intersection  will  give  the  curve  line,  d  g,  which  will  be  the 
one  required  for  the  edge  of  the  soffit.  The  other  edge,  c  h,  is 
found  in  the  same  manner. 

448. —  To  find  the  form  of  the  soffit  for  circular  window- 
heads,  when  the  face  of  the  wall  is  curved.  Let  abed,  (Fig. 
284,)  be  the  ground-plan  of  a  given  window,  and  ef  a,  a  vertical 
section  of  the  head  taken  at  right  angles  to  the  face  of  tho  jambs. 


328 


AMERICAN    HOUSE-CARPENTER. 


Fig.  284. 


Proceed  as  in  the  foregoing  article  to  obtain  the  line,  d  g;  theu 
that  will  be  the  curve  required  for  the  edge  of  the  soffit;  the 
other  edge  being  found  in  the  same  manner. 

If  the  given  vertical  section  be  taken  in  a  line  with  the  face  of 
the  wall,  instead  of  at  right  angles  to  the  face  of  the  jambs,  place 
it  upon  the  line,  c  b,  (Fig.  283 ;)  and,  having  drawn  ordinates  at 
right  angles  to  c  6,  transfer  them  to  ef  a  ;  in  this  way,  a  section 
at  right  angles  to  the  jambs  can  be  obtained. 


SECTION  VL— STAIRS. 


449. — The  STAIRS  is  that  mechanical  arrangement  in  a  build- 
ing by  which  access  is  obtained  from  one  story  to  another.  Theii 
position,  form  and  finish,  when  determined  with  discriminating 
taste,  add  greatly  to  the  comfort  and  elegance  of  a  structure.  As 
regards  their  position,  the  first  object  should  be  to  have  them  near 
the  middle  of  the  building,  in  order  that  an  equally  easy  access 
may  be  obtained  from  all  the  rooms  and  passages.  Next  in  im- 
portance is  light ;  to  obtain  which  they  would  seem  to  be  best 
situated  near  an  outer  wall,  in  which  windows  might  be  construc- 
ted for  the  purpose  ;  yet  a  sky-light,  or  opening  in  the  roof,  would 
-not  only  provide  light,  and  so  secure  a  central  position  for  the 
stairs,  but  may  be  made,  also,  to  assist  materially  as  an  ornament 
to  the  building,  and,  what  is  of  more  importance,  afford  an  op- 
portunity for  better  ventilation. 

450. — It  would  seem  that  the  length  of  the  raking  side  of  the 
pitch-board,  or  the  distance  from  the  top  of  one  riser  to  the  top  ot 
the  next,  should  be  about  the  same  in  all  cases  ;  for,  whether  stairs 
be  intended  for  large  buildings  or  for  small,  for  public  or  for  pri- 
vate, the  accommodation  of  men  of  the  same  stature  is  to  be  con- 
sulted in  every  instance.  But  it  is  evident  that,  with  the  same 
effort,  a  longer  step  can  be  taken  on  level  than  on  rising  ground 
42 


330 


AMERICAN    HOUSE-CARPENTER. 


and  that,  although  the  tread  and  rise  cannot  be  proportioned 
merely  in  accordance  with  the  style  and  importance  of  the  build- 
ing, yet  this  may  be  done  according  to  the  angle  at  which  the 
flight  rises.  If  it  is  required  to  ascend  gradually  and  easy,  the 
length  from  the  top  of  one  rise  to  that  of  another,  or  the  hypothe 
nuse  of  the  pitch-board,  may  be  long  ;  but,  if  the  flight  is  steep 
the  length  must  be  shorter.  Upon  this  data  the  following  pi  oblerr 
is  constructed. 


451. —  To  proportion  the  rise  and  tread  to  one  another, 
Make  the  line,  a  b,  (Fig.  285,)  equal  to  24  inches  ;  from  b,  erect 
b  c,  at  right  angles  to  a  b,  and  make  b  c  equal  to  12  inches  ;  join  a 
and  c,  and  the  triangle,  a  b  c,  will  form  a  scale  upon  which  tc 
graduate  the  sides  of  the  pitch-board.  For  example,  suppose  a 
very  easy  stairs  is  required,  and  the  tread  is  fixed  at  14  inches. 
Place  it  from  b  to/,  and  from/;  draw/^,  at  right  angles  to  a  b  ; 
then  the  length  of  f  g  will  be  found  to  be  5  inches,  which  is  a 
proper  rise  for  14  inches  tread,  and  the  angle,  f  b  g,  will  show 
the  degree  of  inclination  at  which  the  flight  will  ascend.  But,  in 
a  majority  of  instances,  the  height  of  a  story  is  fixed,  while  the 
length  of  tread,  or  the  space  that  the  stairs  occupy  on  the  lower 
floor,  is  optional.  The  height  of  a  story  being  determined,  the 
height  of  each  rise  will  of  course  depend  upon  the  number  intc 
which  the  whole  height  is  divided  ;  the  angle  of  ascent  being  more 
easy  if  the'number  be  great,  than  if  it  be  smaller.  By  dividing 


STAIRS.  331 

the  whole  height  of  a  story  into  a  certain  number  of  rises,  sup- 
pose the  length  of  each  is  found  to  be  6  inches.  Place  this  length 
from  b  to  A,  and  draw  h  i,  parallel  to  a  b  ;  then  h  i,  or  b  j  will  be 
the  proper  tread  for  that  rise,  and  j  b  i  will  show  the  angle  of  as- 
cent. On  the  other  hand,  if  the  angle  of -ascent  be  given,  as  a 
b  I,  (b  I  being  10^  inches,  the  proper  length  of  run  for  a  step- 
ladder,)  drop  the  perpendicular,  I  kt  from  /  to  k  ;  then  I  k  b  will 
be  the  proper  proportion  for  the  sides  of  a  pitch-board  for  that 
run. 

452. — The  angle  of  ascent  will  vary  according  to  circum- 
stances. The  following  treads  will  determine  about  the  right  in* 
clination  for  the  different  classes  of  buildings  specified. 

In  public  edifices,  tread  about  14    inches. 

In  first-class  dwellings  "         12£     " 

In  second-class  "  "         11       « 

In  third-class      "  and  cottages    "  9       " 

Step-ladders  to  ascend  to  scuttles,  <fcc.,  should  have  from  10  to 
11  inches  run  on  the  rake  of  the  string.     (See  notes  at  Art.  103." 
453. — The  length  of  the  steps  is  regulated  according  to  the  ex- 
tent and  importance  of  the  building  in  which  they  are  placed, 
-varying  from  3  to  12  feet,  and  sometimes  longer.     Where  two  per- 
sons are  expected  to  pass  each  other  conveniently,  the  shortest 
length  that  will  admit  of  it  is  3  feet ;  still,  in  crowded  cities  where 
land  is  so  valuable,  the  space  allowed  for   passages  being  very 
small,  they  are  frequently  executed  at  2\  feet. 

454. —  To  find  the  dimensions  of  the  pitch-board.  The  first 
.  thing  in  commencing  to  build  a  stairs,  is  to  make  the^z7c/i-board  ; 
this  is  done  in  the  following  manner.  Obtain  very  accurately,  in 
feet  and  inches,  the  perpendicular  height  of  the  story  in  which 
the  stairs  are  to  be  placed.  This  must  be  taken  from  the  top  ol 
the  floor  in  the  lower  stoiy  to  the  top  of  the  floor  in  the  upper 
story.  Then,  to  obtain  the  number  of  rises,  the  height  in  inches 
thus  obtained  must  be  divided  by  5,  6,  7,  8,  or  9,  according  to  the 
quality  and  style  of  the  building  in  which  the  stairs  are  to  be 


332  AMERICAN    HOUSE-CARPENTER. 

built.  For  instance,  suppose  the  building  to  be  a  fiist-class 
dwelling,  and  the  height  ascertained  is  13  feet  4  inches,  or  160 
inches.  The  proper  rise  for  a  stairs  in  a  house  of  this  class  is 
about  6  inches.  Then,  160  divided  by  6,  gives  26§  inches.  This 
being  nearer  27  than  26,  the  number  of  risers,  should  be  27. 
Then  divide  the  height,  160  inches,  by  27,  and  the  quotient  will 
give  the  height  of  one  rise.  On  performing  this  operation,  the 
quotient  will  be  found  to  be  5  inches,  |  and  T'7  of  an  inch. 

Then,  if  the  space  for  the  extension  of  the  stairs  is  not  limited, 
the  tread  can  be  found  as  at  Art.  451.  But,  if  the  contrary  is  the 
case,  the  whole  distance  given  for  the  treads  must  be  divided  by 
the  number  of  treads  required.  On  account  of  the  upper  floor 
forming  a  step  for  the  last  riser,  the  number  of  treads  is  always 
one  less  than  the  number  of  risers.  Having  obtained  this 
rise  and  tread,  the  pitch-board  may  be  made  in  the  follow- 
ing manner.  Upon  a  piece  of  well-seasoned  board  about  |  of  an 
inch  thick,  having  one  edge  jointed  straight  and  square,  lay  the 
corner  of  a  carpenters'-square,  as  shown  at  Fig.  286.  Make  a  b 


Fig  286. 


equal  to  the  rise,  and  b  c  equal  to  the  tread  ;  mark  along  those 
edges  with  a  knife,  and  cut  it  out  by  the  marks,  making  the  edges 
perfectly  square.  The  grain  of  the  wood  must  run  in  the  direction 
indicated  in  the  figure,  because,  if  it  shrinks  a  trifle,  the  rise  and 
the  tread  will  be  equally  affected  by  it.  When  a  pitch-board  is 
first  made,  the  dimensions  of  the  rise  and  tread  should  be  pre- 
served in  figures,  in  order  that,  should  the  first  shrink,  a  second 
could  be  made. 
455.  —  To  lay  out  the  string.  The  space  required  for  timbei 


STAIRS. 

d  e  f 


333 


and  plastering  under  the  steps,  is  about  5  inches  for  ordinary 
stairs  ;  set  a  gauge,  therefore,  at  5  inches,  and  run  it  on  the  lowei 
edge  of  the  plank,  as  a  6,  (Fig.  287.)  Commencing  at  one  end, 
lay  the  longest  side  of  the  pitch-board  against  the  gauge-mark,  a 
b,  as  at  c,  and  draw  by  the  edges  the  lines  for  the  first  rise  and 
tread;  then  place  it  successively  as  at  d,  e  and/,  until  the  re- 
quired number  of  risers  shall  be  laid  down. 


k 


Fig.  2S8. 

456. — Fig.  288  represents  a  section  of  a  step  and  riser,  joined 
after  the  most  approved  method.  In  this,  a  represents  the  end  of 
a  block  about  2  inches  long,  two  of  which  are  glued  in  the  corner 
in  the  length  of  the  step.  The  cove  at  b  is  planed  up  square, 
glued  in,  and  stuck  after  the  glue  is  set. 


PLATFORM    STAIRS. 

457. — A  platform  stairs  ascends  from  one  story  to  another  in 
two  or  more  flights,  having  platforms  between  for  resting  and 
to  cnange  their  direction.  This  kind  of  stairs  is  the  most  easily 
constructed,  and  is  therefore  the  most  common.  The  cylin- 


334 


AMERICAN    HOUSE-CARPENTER, 


der  is  generally  of  small  diameter,  in  most  cases  about  6  inches. 
It  may  be  worked  out  of  one  solid  piece,  but  a  better  way  is  to 
glue  together  three  pieces,  as  in  Fig-  289;  in  which  the  pieces, 
a,  b  and  c,  compose  the  cylinder,  and  d  and  e  represent  parts  of 
the  strings.  The  strings,  after  being  glued  to  the  cylinder,  are 
secured  with  screws.  The  joining  at  o  and  o  is  the  most  proper 
for  that  kind  of  joint. 

458. —  To  obtain  the  form  of  the  lower  edge  of  the  cylinder . 
Find  the  stretch-out,  d  e,  (Fig.  290,)  of  the  face  of  the  cylinder 
a  b  c,  according  to  Art.  92  ;  from  d  and  e,  draw  d  f  and  e  g,  at 
right  angles  {ode;  draw  h  g,  parallel  to  d  e,  and  make  hf  and 
g  «,  each  equal  to  one  rise;  from  i  and/,  draw  ij  and/ A:,  paral- 
lel to  h  g ;  place  the  tread  of  the  pitch-board  at  these  last  lines, 
and  draw  by  the  lower  edge  the  lines,  k  h  and  i  I ;  parallel  to 
these,  draw  m  n  and  o  p,  at  the  requisite  distance  for  the  dimen- 
sions of  the  string ;  from  s,  the  centre  of  the  plan,  draw  5  q. 
parallel  to  df;  divide  h  qand  q  g,  each  into  2  equal  parts,  as  at 
v  and  w  ;  from  v  and  w,  draw  v  n  and  w  o,  parallel  iofd;  join  n 
and  0,  cutting  q  s  in  r  ;  then  the  angles,  u  n  r  and  rot.  being 
eased  off  according  to  Art.  89,  will  give  the  proper  curve  for  the 
bottom  edge  of  the  cylinder.  A  centre  may  be  found  upon  which 
to  describe  these  curves  thus  :  from  u,  draw  u  x,  at  right  angles 
to  m  n  ;  irom  r,  draw  r  v,  at  right  angles  to  n  o  ;  then  x  will  be 
the  centre  for  the  curve,  u  r.  The  centre  for  the  ?,urve,  r  t,  is 
found  in  the  same  manner. 


STAIRS. 


335 


Fig.  290. 

459. —  To  find  the  position  for  the  balusters.  Place  the 
centre  of  the  first  baluster,  (b.  Fig.  291,)  \  its  diameter  from  the 
face  of  the  riser,  c  d,  and  i  its  diameter  from  the  end  of  the  step, 
e  d  ;  and  place  the  centre  of  the  other  baluster,  a,  half  the  treaci 
from  the  centre  of  the  first.  The  centre  of  the  rail  must  be  placed 
over  the  centre  of  the  balusters.  Their  usual  length  is  2  feet 
5  inches,  and  2  feet  9  inches,  for  the  short  and  the  long  balusters 
respectively. 


Fig.  291. 


330 


AMERICAN    HOUSE-CARPENTER. 


460. —  To  find  the  face-mould  for  a  round  hand-rail  to  plat" 
form  stairs.  CASE  1. —  When  the  cylinder  is  small.  In  Fig. 
292,  j  and  e  represent  a  vertical  section  of  the  last  two  steps  of  the 
first  flight,  and  d  and  i  the  first  two  steps  of  the  second  flight,  ot 
a  platform  stairs,  the  line,  e  f,  being  the  platform  ;  and  a  b  c  is 
the  plan  of  a  line  passing  through  the  centre  of  the  rail  around 
the  cylinder.  Through  i  and  </,  draw  i  k,  and  through  j  and  e, 
draw^  k  ;  from  k,  draw  k  I,  parallel  to  f  e  ;  from  6,  draw  b  m, 
parallel  tog  d;  from  I,  draw  I  r,  parallel  to  kj  ;  from  w,  draw  n 
£,  at  right  angles  to^'  k  ;  on  the  line,  o  b,  make  a  t  equal  to  n  t ; 
join  c  and  t :  on  the  line,  j  c,  (Fig.  293,)  make  e  c  equal  to  e  n  at 
Fig.  292  ;  from  c,  draw  c  t,  at  right  angles  toj  c,  and  make  c  t 


327 


Fig.  293. 


equal  to  c  t  at  Fig.  292  ;  through  t,  draw  p  I,  parallel  toj  e,  and 
make  1 1  equal  to  1 1  at  Fig.  292  ;  join  I  and  c,  and  complete  the 
parallelogram,  eels;  find  the  points,  o,  o,  o,  according  to  Art. 
118  ;  upon  e,  o,  o,  o,  and  J.  successively,  with  a  radius  equal  to 
half  the  width  of  the  rail,  describe  the  circles  shown  in  the  figure  ; 
then  a  curve  traced  on  both  sides  of  these  circles  and  just  touch- 
ing them,  will  give  the  proper  form  for  the  mould.  The  joint  at 
I  is  drawn  at  right  angles  to  c  I. 

461f — Elucidation  of  the  foregoing  method.  This  excellent 
plan  for  obtaining  the  face-moulds  for  the  hand-rail  of  a  platform 
stairs,  has  never  before  been  published.  It  was  communicated  to 
me  by  an  eminent  stair-builder  of  this  city :  and  having  seen 
rails  put  up  from  it,  I  am  enabled  to  give  it  my  unqualified  re- 
commendation. In  order  to  have  it  fully  understood,  I  have  in- 
troduced Fig.  294 ;  in  which  the  cylinder,  for  this  purpose,  is 
made  rectangular  instead  of  circular.  The  figure  gives  a  per- 
spective view  of  a  part  of  the  upper  and  of  the  lower  flights,  and 
a  part  of  the  platform  about  the  cylinder.  The  heavy  lines,  i  m, 
in  c  and  c  ;',  show  the  direction  of  the  rail,  and  are  supposed  to 
pass  through  the  centre  of  .it.  When  the  rake  of  the  second 
flight  is  the  same  as  that  of  the  first,  whicl.  is  here  and  is  gene- 
rally the  case,  the  face-mould  for  the  lower  twist  will,  when  re- 
versed, do  for  the  upper  flight:  that  part  of  the  rail,  therefore, 
which  passes  from  e  to  c  and  from  c  to  /,  is  all  that  will  need  ex- 
planation. 

Suppose,  then,  that  the  parallelogram,  e  a  o  c,  represent  a  plane 
lying  perpendicularly  over  e  a  bf,  being  inclined  in  the  direction, 
e  c,  and  level  in  the  direction,  c  o ;  suppose  this  plane,  e  a  o  c, 
43 


338 


AMERICAN    HOUSE-CARPENTEB. 


Fig.  294 

be  revolved  on  e  c  as  an  axis,  in  the  manner  indicated  by  the  arcs, 
o  n  and  a  x,  until  it  coincides  with  the  plane,  e  r  t  c  ;  the  line,  a 
o,  will  then  be  represented  by  the  line,  x  n  ;  then  add  the  paral- 
lelogram, x  r  t  w,  and  the  triangle,  c  tl,  deducting  the  triangle,  e  r  s : 
and  the  edges  of  the  plane,  e  s  I  c,  inclined  in  the  direction,  ec,  and 
also  in  the  direction,  c  I,  will  lie  perpendicularly  over  the  plane,  e 
abf.  From  this  we  gather  that  t  he  line,  co,  being  at  light  angles  to 


STAIRS.  339 

e  c,  must,  in  order  to  reach  the  point,  /,  be  lengthened  the  distance, 
n  t,  and  the  right  angle,  e  c  t,  be  made  obtuse  by  the  addition  to 
it  of  the  angle,  t  c  I.  By  reference  to  Fig.  292,  it  will  be  seen 
that  this  lengthening  is  performed  by  forming  the  right-angled 
triangle,  cot,  corresponding  to  the  triangle,  c  o  t,  in  Fig.  294. 
The  line,  c  £,  is  then  transferred  to  Fig.  293,  and  placed  at  right 
angles  to  e  c ;  this  angle,  e  c  t,  being  increased  by  adding  the  an- 
gle, t  c  I,  corresponding  to  t  c  I,  Fig.  294,  the  point,  /,  is  reached, 
and  the  proper  position  and  length  of  the  lines,  e  c  and  c  I  ob- 
tained. To  obtain  the  face-mould  for  a  rail  over  a  cylindrica1 
well-hole,  the  same  process  is  necessary  to  be  followed  until  the 
the  length  and  position  of  these  lines  are  found ;  then,  by  forming 
the  parallelogram,  eels,  and  describing  a  quarter  of  an  ellipse 
therein,  the  proper  form  will  be  given. 


Fig  295. 


462.— CASE  2  —  When  the  cylinder  is  large.     Ftp.  295  re- 


840 


AMERICAN    HOUSE-CARPENTER, 


presents  a  plan  and  a  vertical  section  of  a  line  passing  through  tha 
centre  of  the  rail  as  before.  From  b,  draw  b  A;,  parallel  to  c  d  ;  ex- 
tend the  lines,  i  d  and  j  e,  until  they  meet  kb'mk  and/;  from  n, 
draw  n  I,  parallel  to  o  b  ;  through  I,  draw  1 t,  parallel  tojk,  from 
k,  draw  k  t,  at  right  angles  to  j  k  ;  on  the  line,  o  6,  make  o  t  equal 
to  k  t.  Make  e  c,  (Fig.  296.)  equal  toe  k  at  Fig.  295  ;  from  c, 


Fig.  296. 

draw  c  t,  at  right  angles  to  e  c,  and  equal  to  c  /  at  Fig.  295  •  from 
t,  draw  t  p,  parallel  to  c  e,  and  make  1 1  equal  to  1 1  at  Fig.  295  ; 
complete  the  parallelogram,  eels,  and  find  the  points,  o,  o,  o,  as 
before ;  then  describe  the  circles  and  complete  the  mould  as  in 
Fig.  293  The  difference  between  this  and  Case  1  is,  that  the 
line,  c  t,  instead  of  being  raised  and  thrown  out,  is  lowered  and 
drawn  in.  (See  note  at  page  381.) 


463. — CASE  3. —  Where  the  rake  meets  the  level.     In  Fig 


STAIRS.  341 

297,  a  b  c  is  the  plan  of  aline  passing  through  the  centre  of  tne 
rail  around  the  cylinder  as  before,  and  j  and  e  is  a  vertical  section 
of  two  steps  starting  from  the  floor,  h  g.  Bisect  e  h  in  «f,  and 
through  d,  draw  df,  parallel  to  h  g  ;  bisect/  n  in  /,  and  trora  I, 
draw  /  t,  parallel  to  nj;  from  n,  draw  n  t,  at  right  angles  tojn, 
on  the  line,  o  b,  make  o  t  equal  to  n  t.  Then,  to  obtain  a  mould 
for  the  twist  going  up  the  flight,  proceed  as  at  Fig.  293  ;  making 
e  r.  in  that  figure  equal  to  e  n  in  Fig.  297,  and  the  other  lines  of 
a  length  and  position  such  as  is  indicated  by  the  letters  of  reference 
in  each  figure.  To  obtain  the  mould  for  the  level  rail,  extend  h 
0,  (Fig.  297,)  to  i  ;  make  o  i  equal  to/  /,  and  join  i  and  c  ;  maVe 
c  t,  (Fig.  298,)  equal  to  c  i  at  Fig.  297 ;  through  c,  dra.w  c  '/,  at 


right  angles  to  c  i  ;  make  d  c  equal  to  df  at  Fig.  297,  and  com 
plete  the  parallelogram,  o  d  c  i;  then  proceed  as  in  the  previous 
cases  to  find  the  mould. 

464. — All  the  moulds  obtained  by  the  preceding  examples  have 
been  for  round  rails.  For  these,  the  mould  may  be  applied  to 
a  plank  of  the  same  thickness  as  the  rail  is  intended  to  be,  and 
the  plank  sawed  square  through,  the  joints  being  cut  square  from 
the  face  of  the  plank.  A  twist  thus  cut  and  truly  rounded  will 
hang  in  a  proper  position  over  the  plan,  and  present  a  perfect  and 
graceful  wreath. 

465. —  To  bore  for  the  balusters  of  a  round  rail  before  ro  md- 
ing  it.  Make  the  angle,  o  c  t,  (Fig.  299,)  equal  to  the  angle,  o 
c  t,  at  Fig.  292  ;  upon  c,  describe  a  circle  with  a  radius  equal  to 
half  the  thickness  of  the  rail ;  draw  the  tangent,  b  d,  parallel  to 
t  c,  and  complete  the  rectangle,  e  b  df,  having  sides  tangical  to 
the  circle;  from  c,  draw  c  a,  at  right  angles  to  oc;  then,  b  d 
being  the  bottom  of  the  rai'l,  set  a  gauge  from  b  to  a,  and  run  it 
the  whole  length  of  the  stuff;  in  boring,  place  the  centre  of  tb.3 


AMERICAN    E30USE-CARPENTER. 


bit  in  the  gauge-mark  at  a,  and  bore  in  the  direction,  a  c.  To  do 
this  easily,  make  chucks  as  represented  in  the  figure,  the  bottom 
edge,  g  h,  being  parallel  to  o  c,  and  having  a  place  sawed  out,  as 
ef,  to  receive  the  rail.  These  being  nailed  to  the  bench,  the  rail 
will  be  held  steadily  in  its  proper  place  for  boring  vertically, 
The  distance  apart  that  the  balusters  require  to  be,  on  the  under 
side  of  the  rail,  is  one-half  the  length  of  the  rake-side  of  the 
pitch -board. 


STAIRS.  343 

466. — To  obtain,  by  the  foregoing  principles,  the  face-mould 
for  the  twists  of  a  moulded  rail  upon  platform  stairs  In  Fig. 
300,  a  b  c  is  the  plan  of  a  line  passing  through  the  centre  of 
the  rail  around  the  cylinder  as  before,  and  the  lines  above 
it  are  a  vertical  section  of  steps,  risers  and  platform,  with 
the  lines  for  the  rail  obtained  as  in  Fig.  292.  Set  half  the  width 
of  the  rail  from  b  to  f  and  from  b  to  r,  and  from  /  and  r,  draw/ 
e  and  r  d,  parallel  to  c  a  At  Fig.  301,  the  centre  lines  of  the 

s       d      n    e 


g 

Fig.  301. 

rai",  k  c  and  c  n,  are  obtained  as  in  the  previous  examples.  Make 
c  i  and  c  j,  each  equal  to  c  i  at -Fig.  300,  and  draw  the  lines,  i  m 
and  j  g,  parallel  to  c  k  ;  make  n  e  and  n  d  equal  to  n  e  and  n  cZ  at 
Fzg-.  300,  and  draw  d  o  and  e  I,  parallel  to  n  c ;  also,  through  k, 
draw  s  g,  parallel  to  n  c ;  then,  in  the  parallelograms,  m  s  d  o  and 
g  s  el,  find  the  elliptic  curves,  d  m  and  e  g,  according  to  Art. 
118,  and  they  will  define  the  curves.  The- line,  d  e,  being  drawn 
through  n  parallel  to  k  c,  defines  the  joint,  which  is  to  be  cut 
through  the  plank  vertically.  If  the  rail  crosses  the  platform  rathei 
steep,  a  butt  joint  will  be  preferable,  to  obtain  wb:ch  see  Art.  498. 


344:  AMERICAN    HOUSE-CARPENTEUl. 

467. —  To  apply  the  mould  to  the  plank.  The  mould  obtained 
according  to  the  last  article  must  be  applied  to  both  sides  of  the 
plank,  as  shown  at  Fig.  302.  Before  applying  the  mould,  the 
edge,  e/,  must  be  bevilled  according  to  the  angle,  c  t  #,  at  Fig 
300  ;  if  the  rail  is  to  be  canted  up.  the  edge  must  be  bevilled  at 
an  obtuse  angle  with  the  upper  face ;  but  if  it  is  to  be  canted 
down,  the  angle  that  the  edge  makes  with  the  upper  face  mnst  be 
acute.  From  the  spring  of  the  curve,  a,  and  the  end,  c,  draw 
vertical  lines  across  the  edge  of  the  plank  by  applying  the  pitch- 
board,  a  b  c  ;  then,  in  applying  the  mould  to  the  other  side,  place 
the  points,  a  arid  c"  at  b  and/;  and,  after  marking  around  it,  saw 
the  rail  out  vertically.  After  the  rail  is  sawed  out,  the  bottom 
and  the  top  surfaces  must  be  squared  from  the  sides. 

4G8. —  To  ascertain  the  thickness  of  stuff  required  for  the 
twists.  The  thickness  of  stuff  required  for  the  twists  of  a  round 
rail,  as  before  observed,  is  the  same  as  that  for  the  straight ;  but 
for  a  moulded  rail,  the  stuff  for  the  twists  rnust  be  thicker  than 
that  for  tho  straight.  In  Fig.  300,  draw  a  section  of  the  rail  be- 
tween the  lines,  d  r  and  ef:  and  as  close  to  the  line,  d  e,  as  possi- 
ble ;  at  the  lower  corner  of  the  section,  draw g  A,  parallel  to  d  e; 
then  the  distance  that  these  lines  are  apart,  will  be  the  thickness 
required  for  the  twists  of  a  moulded  rail. 

The  foregoing  method  of  finding  moulds  for  rails  is  applicable 
to  all  stairs  which  have  continued  rails  around  cylinders,  and  are 
without  winders. 

WINDING    STAIRS. 

469. — Winding  stairs  have  steps  tapering  narrower  at  one  end 
than  at  the  other.  In  some  stairs,  there  are  steps  of  parallel  width 
incorporated  with  tapering  steps  ;  the  former  are  then  called  flyers 
and  the  latter  winders. 

470. —  To  describe  a  regular  geometrical  winding  stairs. 
In  Fig.  303,  abed  represents  the  inner  surface  of  the  wall  en^ 
closing  the  space  allotted  to  the  stairs,  a  e  the  length  of  the  steps, 
and  efgh  the  cylinder,  or  face  of  the  front  string.  The  line, 


STAIRS. 


345 


Fig.  808 

a,  e,  is  given  as  the  face  of  the  first  riser,  and  the  point,  j,  for  the 
lirai;  of  the  last.  Make  e  i  equal  to  18  inches,  and  upon  o,  with 
o  i  for  radius,  describe  the  arc,  ij;  obtain  the  number  of  risers 
and  of  treads  required  to  ascend  to  the  floor  at  j,  according  to  Art. 
•ioi,  and  divide  the  arc,  ij,  into  the  same  number  of  equal  parts 
as  there  are  to  be  treads  ;  through  the  points  of  division,  1,  2,  3, 
£c.,  and  from  the  wall-string  to  the  front-string,  draw  lines  tend- 
ing to  the  centre,  o  ;  then  these  lines  will  represent  the  face  ot 
each  riser,  and  determine  the  form  and  width  of  the  steps.  Allow 
the  necessary  projection  for  the  nosing  beyond  a  e,  which  should 
be  equal  to  the  thickness  of  the  step,  and  then  a  elk  will  be  the 
dimensions  for  each  step.  Make  a  pitch-board  for  the  wall-string 
having  a  k  for  the  tread,  and  the  rise  as  previously  ascertained  ; 
with  this,  lay  out  on  a  thicknessed  plank  the  several  risers  and 
treads,  as  at  Fig.  287,  gauging  from  the  upper  edge  of  the  strirg 
for  the  line  at  which  to  set  the  pitch-board. 

Upon  the  back  of  the  string,  with  a  1  ^  inch  dado  plane,  make 

4:4: 


846  AMERICAN    HOUSE-CARPENTER. 

a  succession  of  grooves  1£  inches  apart,  and  parallel  with  the 
lines  for  the  risers  on  the  face.  These  grooves  must  be  »ut  along 
the  whole  length  of  the  plank,  and  deep  enough  to  admit  of  the 
plank's  bending  around  the  curve,  abed.  Then  construct  a 
drum,  or  cylinder,  of  any  common  kind  of  stuff,  and  made  to  fit 
a  curve  having  a  radius  the  thickness  of  the  string  less  than  o  a  ; 
upon  this  the  string  must  be  bent,  and  the  grooves  filled  with  strips 
of  wood,  called  keys,  which  must  be  very  nicely  fitted  and  glued 
in.  After  it  has  dried,  a  board  thin  enough  to  bend  around  on  the 
outside  of  the  string,  must  be  glued  on  from  one  end  to  the  other 
and  nailed  with  clout  nails.  In  doing  this,  be  careful  not  to  nail 
into  any  place  where  a  riser  or  step  is  to  enter  on  the  face. 

After  the  string  has  been  on  the  drum  a  sufficient  time  for  the 
glue  to  set,  take  it  off,  and  cut  the  mortices  for  the  steps  and 
risers  on  the  face  at  the  lines  previously  made  ;  which  may  be 
done  by  boring  with  a  centre-bit  half  through  the  string,  and 
nicely  chiseling  to  the  line.  The  drum  need  not  be  made  so 
large  as  the  whole  space  occupied  by  the  stairs,  but  merely  large 
enough  to  receive  one  piece  of  the  wall-string  at  once  —  for  it 
is  evident  that  more  than  one  will  be  required.  The  front  string 
may  be  constructed  in  the  same  manner  ;  taking  e  I  instead  of  a 
k  for  the  tread  of  the  pitch-board,  dadoing  it  with  a  smaller  dado 
plane,  and  bending  it  on  a  drum  of  the  proper  size. 


Fig.  804. 

471.  —  To  find  the  shape  and  position  of  the  timbers  neces- 
sary to  support  a  winding  stairs.  The  dotted  lines  in  Fig. 
303  show  the  proper  position  of  the  timbers  as  regards  the  plan  : 
the  shape  of  each  is  obtained  as  follows.  In  Fig.  304,  the  line, 
1  a,  is  equal  to  a  riser,  less  the  thickness  of  the  floor,  and  the 
»ines,  2  w,  3  n,  4  o,  5  p  and  6  7,  are  each  equal  to  one  riser.  The 


STAIRS.  347 

line,  a  2,  is  equal  to  a  in  in  Fig.  303,  the  line,  m  3  to  #i  w  in  that 
figure,  (fee.  In  drawing  this  figure,  commence  at  a,  and  make 
the  lines,  a  1  and  a  2,  of  the  length  above  specified,  and  drau 
them  at  right  angles  to  each  other ;  draw  2  ra,  at  right  angles  to 
a  2,  and  m  3,  at  right  angles  to  m  2,  and  make  2  m  and  m  3  of 
the  lengths  as  above  specified  ;  and  so  proceed  to  the  end.  Then, 
through  the  points,  1,  2,  3,  4,  5  and  6.  trace  the  line,  1  b ;  upon 
the  points,  1,  2,  3,  4,  &c.,  with  the  size  of  the  timber  for  radius, 
describe  arcs  as  shown  in  the  figure,  and  by  these  the  lower  line 
may  be  traced  parallel  to  the  upper.  This  will  give  the  proper 
shape  for  the  timber,  a  6,  in  Fig.  303  ;  and  that  of  the  others  may 
be  found  in  the  same  manner.  In  ordinary  cases,  the  shape  of 
one  face  of  the  timber  will  be  sufficient,  for  a  good  workman 
can  easily  hew  it  to  its  proper  level  by  that ;  but  where  great 
accuracy  is  desirable,  a  pattern  for  the  other  side  may  be  found 
in  the  same  manner  as  for  the  first. 

472. —  To  find  the  falling-mould  for  the  rail  of  a  winding 
stairs.  In  Fig.  305,  a  c  b  represents  the  plan  of  a  rail  around 
half  the  cylinder,  A  the  cap  of  the  newel,  and  1,  2,  3,  &c.,  the 
face  of  the  risers  in  the  order  they  ascend.  Find  the  stretch-out, 
ef,  of  a.c  b,  according  to  Art.  02;  from  o,  through  the  point  of 
the  mitre  at  the  newel-cap,  draw  o  s  ;  obtain  on  the  tangent,  e  rf, 
the  position  of  the  points,  s  and  K2*  as  at  t  and/2 ;  from  e  tf2  and 
fj  draw  ex,  t  M,/2  g2  and  f  h,  all  at  right  angles  to  e  d  ;  make  e 
g  equal  to  one  rise  and  /2  g*  equal  to  12,  as  this  line  is  drawn 
from  the  12th  riser ;  from  g,  through  g*,  drawg-  i;  make -g  x  equal 
to  about  three-fourths  of  a  rise,  (the  top  of  the  newel,  ar,  should 
be  3|  feet  from  the  floor  ;)  draw  x  u,  at  right  angles  to  e  x,  and 
ease  off  the  angle  at  u  ;  at  a  distance  equal  to  the  thickness  of 

*  In  the  above,  the  references,  a2,  &*,  &c.,  are  introduced  for  the  first  time.  During  the 
lime  taken  to  refer  to  the  figure,  the  memory  of  the  form  of  these  may  pass  from  the  mind, 
While  that  of  the  sound  alone  remains ;  they  mi  y  then  be  mistaken  for  a  2,  6  2,  &c.  Thia 
din  be  avoided  in  reading  by  giving  them  a  sound  corresponding  to  their  meaning,  which 
ta  tecond  a  second  b,  &c.  or  a  second,  b  second. 


348 


AMERICAN    HOUSE-CARPENTER. 


Fig.  805. 

the  rail,  draw  v  w  y,  parallel  to  x  u  i;  from  the  centre  of  the  plan, 
o,  draw  o  l:  at  right  angles  to  e  d  ;  bisect  h  n  in  p,  and  through 
7?,  at  right  angles  tog-  i,  draw  a  line  for  the  joint ;  in  the  same 
manner,  draw  the  joint  at  k  ;  then  x  y  will  be  the  falling-mould 
for  that  part  of  the  rail  which  extends  from  s  to  b  on  the  plan. 

473. —  To  find  tJ  .e  face-mould  for  t.Jie  rail  of  a  winding-stairs. 
From  the  extremities  of  the  joints  in  the  falling-mould,  as  A*,  z 
and  y,  (Fig.  305,)  draw  k  a2,  z  bz  and  y  d:  at  right  angles  to  e  d  ; 
make  b  e2  equal  to  /  d.  Then,  to  obtain  the  direction  of  the 
joint,  a2  c2,  or  62  d\  proceed  as  at  Fig  306,  at  which  the  parts  are 


349 


Fig.  306. 


shown  at  half  their  full  size.  A  is  the  plan  of  the  rail,  and  B  is 
the  falling-mdiikl  :  in  which  k  z  is  the  direction  of  the  butt-joint. 
From  k,  draw  k  b,  parallel  to  I  o,  and  k  e,  at  right  angles  to  k  b  : 
from  6,  draw  b  /,  tending  to  the  centre  of  the  plan,  and  from/,  draw 
/  e,  parallel  to  b  k  ;  from  /,  through  6  draw  I  i,  and  from «,  draw  i 
d,  parallel  to  ef;  join  d  and  b,  and  d  b  will  be  the  proper  direction 


350 


AMERICAN    HOUSE-CARPENTER. 


for  the  joint  on  the  plan.  The  direction  of  the  joint  on  the  othei 
side,  a  c,  can  be  found  by  transferring  the  distances,  x  b  arid  o  d 
to  x  a  and  o  c.  (See  Art.  477.) 


Fig.   307. 


Having  obtained  the  direction  of  the  joint,  make  s  r  d  b,  (Fig. 
307,)  equal  to  s  r  d  62  in  Fig.  305  ;  through  r  and  </,  draw  t  a  e 
through  6-  and  from  d,  draw  t  u  and  d  e,  at  right  angles  to  /  a  , 
make  t  u  and  d  e  equal  to  t  u  and  6*  m,  respectively,  in  Fig.  305  ; 
from  u,  through  e,  draw  u  o  ;  through  6,  from  r,  and  from  as  many 
other  points  in  the  line,  /  a,  as  is  thought  necessary,  as/,  h  and  ; 
draw  the  ordinates,  r  c,  f  g,  h  i,j  k  and  a  o  ;  from  u,  c,  g,  i,  k,  e 
and  o,  draw  the  ordinates,  u  1,  c  2.  g  3,  i  4,  k  5,  e  6  and  o  7,  at 
right  angles  to  u  o  ;  make  M  1  equal  to  £  5,  c  2  equal  to  r  2,  £•  3 
equal  to/  3,  &c.,  and  trace  the  curve,  1  7,  through  the  points 
thus  found  ;  find  the  curve,  c  e,  in  the  same  manner,  by  transfer- 
ring the  distances  between  the  line,  /  a,  and  the  arc,  r  d;  join  1 
and  c,  also  e  and  7 ;  then,  I  c  e  7  will  be  the  face-mould  required 
for  that  part  of  the  rail  which  is  denoted  by  the  letters,  s  r  d2  b\ 
on  the  plan  at  Fig.  305. 

To  ascertain  the  mould  for  the  next  quarter,  make  a  cj  e, 


STAIRS. 


Fig.  303. 

'60S,)  equal  to  a*  c2  j  e*  at  Fig.  305  ;  at  any  convenient  height  on 
the  line,  d  i,  in  that  figure,  draw  q  i\  parallel  to  e  d  ;  through  c 
and/  (Fig.  308,)  draw  b  d  ;  through  a,  and  from  j,  draw  b  k  and 
;  o,  at  right  angles  to  b  d  ;  make  b  k  andj  o  equal  to'i2  k  and  5 
?',  respectively,  in  Fig.  305;  from/:,  through  o,  draw  kf ;  and 
proceed  as  in  the  last  figure  to  obtain  the  face-mould,  A. 

474. —  To  ascertain  the  requisite  thickness  of  stuff.  CASE 
1. —  When  the  falling-mould  is  straight.  Make  o  h  and  k  m, 
(Fig.  308,)  equal  to  i  y  at  Fig.  305  ;  draw  h  i  and  m  n,  parallel 
to  b  d  ;  through  the  corner  farthest  from  kf,  as  n  or  i,  draw  n  i, 
parallel  to  kf;  then  the  distance  between  kf  and  n  i  will  give 
the  thickness  required. 

475. — CASE  2. —  When  the  falling-mould  is  curved.  In  Fig. 
309,  s  r  d  b  is  equal  to  5  r  d2  62  in  Fig.  305.  Make  a  c  equal  to  the 
stretch-out  of  the  arc,  5  b,  according  to  Art.  92,  and  divide  a  c  and 
s  6.  each  into  a  like  number  of  equal  parts  ;  from  a  and  c.  and  from 
each  point  of  division  in  the  line,  a  c,  draw  a  k,  el,  (fee.,  at  right  an- 
gles to  a  c  ,  make  a  A- equal  to  t  u  in  Fig.305,nndcj  equal  to  62wi 


352 


AMERICAN    HOUSE-CARPENTER. 


a      e     f     g      h      i     c 


Fig.  3-:0. 


in  that  figure,  and  complete  thetalling-monld,  kj,  every  way  equal 
to  u  in  in  Fig.  305  ;  from  the  points  of  division  in  the  arc,  sb,  draw 
lines  radiating  towards  the  centre  of  the. circle,  dividing  the  arc. 
r  d,  in  the  same  proportion  as  5  b  is  divided  ;  from  d  and  6,  draw 
d  t  and  b  u,  at  right  angles  to  a  d,  and  from  j  and  v,  draw  j  u  and  v 
w,  at  right  angles  loj  c  ;  then  a:  t  u  w  will  be  a  vertical  projection 
of  the  joint,  d  b.  Supposing  every  radiating  line  across  s  r  d  b — 
corresponding  to  the  vertical  lines  across  kj — to  represent  a  joint, 
find  their  vertical  projection,  as  at  1,  2,  3,  4,  5  and  6  ;  through  the 
corners  of  those  parallelograms,  trace  the  curve  lines  shown  in  th.e 
figure  ;  then  6  u  will  be  a  helinet,  or  vertical  projection,  of  s  r  d  b. 
To  find  the  thickness  of  plank  necessary  to  get  out  this  part  of 
the  rail,  draw  the  line,  z  t,  touching  the  upper  side  of  the  helinet 
in  two  places  :  through  the  corner  farthest  projecting  from  that 
line,  as  w,  draw  y  ic,  parallel  to  z  t ;  then  the  distance  between 
those  lines  will  be  the  proper  thickness  of  stuff  for  this  part  of  the 
rail.  The  same  process  is  necessary  to  find  the  thickness  of 
stuff  in  all  cases  in  which  the  falling-mould  is  in  any  way  curved. 
476. —  To  apply  the  face-mould  to  the  plank.  In  Fig.  310, 
A  represents  the  plank  with  its  best  side  and  edge  in  view,  and 
B  the  same  plunk  turned  up  so  as  to  bring  in  view  the  other  side 


353 


Fig.  810. 


and  the  same  edge,  this  being  square  from  the  face.  Apply  the 
tips  of  the  mould  at  the  edge  of  the  plank,  as  at  a  and  o,  (.4,)  and 
mark  out  the  shape  of  the  twist ;  from  a  and  o,  draw  the  lines,  a 
b  and  o  c,  across  the  edge  of  the  plank,  the  angles,  e  a  b  and  e  o 
c,  corresponding  with  k  fd  at  Fig.  308  ;  turning  the  plank  up  as 
at  jB,  apply  the  tips  of  the  mould  at  b  and  c,  and  mark  it  out  as 
shown  in  the  figure.  In  sawing  out  the  twist,  the  saw  must  be 
be  moved  in  the  direction,  a  b  ;  which  direction  will  be  perpen- 
dicular when  the  twist  is  held  up  in  its  proper  position. 

In  sawing  by  the  face-mould,  the  sides  of  the  rail  are  obtained  ; 
the  top  and  bottom,  or  the  upper  and  the  lower  surfaces,  a're  ob- 
tained by  squaring  from  the  sides,  after  having  bent  the  falling- 
mould  around  the  outer,  or  convex  side,  and  marked  by  its  edges. 
Marking  across  by.  the  ends  of  the  falling-mould  will  give  the 
position  of  the  butt-joint. 

477. — Elucidation  of  the  process  by  which  the  direction  of 
the  butt-joint  is  obtained  in  Art.  473.  Mr.  Nicholson,  in  his 
Carpenter's  Guide,  has  given  the  joint  a  different  direction  to 
that  here  shown  ;  he  radiates  it  towards  the  centre  of  the  cylin- 
der. This  is  erroneous — as  can  be  shown  by  the  following 
operation : 

In  Fig.  311,  a  r  j  i  is  the  plan  of  a  part  of  the  rail  about  the 
joint,  s  uis  the  stretch-out  of  a  i,  and  gp  is  the  helinet,  or  ver- 
tical projection  of  the  plan,  a  r  j  i,  obtained  according  to   Art 
45 


•       AMERICAN    HOIfSE-CARPENTEH. 
g 


i75.  Bisect  r  t,  part  of  an  ordinate  from  the  centre  of  the  plan, 
and  through  the  middle,  draw  c  b,  at  right  angles  to  g  v  ;  from 
b  and  c,  draw  c  d  and  b  e,  at  right  angles  to  s  u  ;  from  d  and  e, 
draw  lines  radiating  towards  the  centre  of  the  plan :  then  d  o 
<md  e  rti  will  be  the  direction  of  the  joint  on  the  plan,  according  to 
Nicholson,  and  c  6  its  direction  on  the  falling-mould.  It  will  be 
admitted  that  all  the  lines  on  the  upper  or  the  lower  side  of  the  rail 
which  radiate  towards  the  centre  of  the  cylinder,  as  d  o,  e  m  or 
i^',  are  level ;  for  instance,  the  level  line,  w  v,  on  the  top  of  the 


STAIRS.  355 

rail  in  the  helinet,  is  a  true  representation  of  the  radiating  line,  j  i} 
on  the  plan.  The  line,  b  h,  therefore,  on  the  top  of  the  rail  in 
the  helinet,  is  a  true  representation  of  e  m  on  the  plan,  and  k  c  on 
the  bottom  of  the  rail  truly  represents  d  o.  From  k,  draw  k  /, 
parallel  to  c  6,  and  from  h,  draw  hf,  parallel  to  b  c;  join  /  and 
6,  also  c  and/;  then  c  k  I  b  will  be  a  true  representation  of  the 
end  of  the  lower  piece,  B,  and  c  fh  b  of  the  end  of  the  upper 
piece,  A ;  and/  k  or  h  I  will  show  how  much  the  joint  is  open  on 
the  inner,  ot  concave  side  of  the  rail. 


356 


AMERICAN    HOUSE  -CARPENTER. 


To  show  that  the  process  followed  in  Art.  473  is  correct,  let  do 
and  e  m,  (Fig.  312,)  be  the  direction  of  the  butt-joint  found  as  at 
Fig.  306.  Now,  to  project,  on  the  top  of  the  rail  in  the  helinet,  a 
line  that  does  not  radiate  towards  the  centre  of  the  cylinder,  as  j 
k,  draw  vertical  lines  from  j  and  k  to  w  and  h,  and  join  w  and  h  ; 
then  it  will  be  evident  that  w  h  is  a  true  representation  in  the  helinet 
of  j  k  on  the  plan,  it  being  in  the  same  plane  as  j  k,  and  also  in  the 
same  winding  surface  as  w  v.  The  Hue,  I  n,  also,  is  a  true  repre- 
sentation on  the  bottom  of  the  helinet  of  the  line,^'  k,  in  the  plan. 
The  line  of  the  joint,  e  m,  therefore,  is  projected  in  the  same  way 
and  truly  by  i  b  on  the  top  of  the  helinet ;  and  the  line,  d  o,  by 
c  a  on  the  bottom.  Join  a  and  i,  and  then  it  will  be  seen  that 
the  lines,  c  a,  a  i  and  i  b,  exactly  coincide  with  c  b,  the  line  of 
he  joint  on  the  convex  side  of  the  rail ;  thus  proving  the  lower 
uid  of  the  upper  piece,  A,  and  the  upper  end  of  the  lower  piece, 
B,  to  be  in  one  and  the  same  plane,  and  that  the  direction  of  the 
joint  on  the  plan  is  the  true  one.  By  reference  to  Fig.  306  it  will 
be  seen  that  the  line,  /  i,  corresponds  to  #  i  in  Fig.  312  ;  and 
that  e  k  in  that  figure  is  a  representation  of/  b,  and  i  k  of  d  b. 


Fig.  813. 


In  getting  out  the  twists,  the  joints,  before  the  falling-mould  is 


STAIRS  357 

applied,  are  cut  perpendicularly,  the  face  inon.d  being  long  enough 
to  include  the  oveiplus  necessary  for  a  butt-joint.  The  face-mould 
for  A,  therefore,  would  have  to  extend  to  the  line,  i  b  ;  and  that  for 
B,  to  the  line,  y  z.  Being  sawed  vertically  at  first,  a  section  of  the 
joint  at  the  end  of  the  face-mould  for  A,  would  be  represented  in 
the  helinet  by  b  ifg-  To  obtain  the  position  of  the  line,  6  i,  on 
the  end  of  the  twist,  draw  i  .<?,  (Fig:  313,)  at  right  angles  to  if, 
and  make  i  s  equal  to  m  e  at  Fig.  312  ;  through  5.  draw  s  g,  pa- 
rallel to  i  /,  and  make  s  b  equal  to  s  b  at  Fig.  312  ;  join  b  and  i  •; 
make  i/equal  to  i  /at  Fig.  312,  and  from  /",  draw  fg,  parallel  to  \ 
b  ;  then  i  b  gf  will  be  a  perpendicular  section  of  the  rail  over  the 
line,  e  m,  on  the  plan  at  Fig.  312,  corresponding  to  i  b  gfm  the 
helinet  at  that  figure  ;  and  when  the  rail  is  squared,  the  top,  or 
back,  must  be  trimmed  off  to  the  line,  i  b,  and  the  bottom  to  the 
\me,fg. 

478. —  To  grade  the  front  string  of  a  stairs,  having  winders 
in  a  quarter-circle  at  the  top  of  the  flight  connected  with  flyers 
at  the  bottom.  In  Fig.  314,  a  b  represents  the  line  of  the  facia 
along  the  floor  of  the  upper  story,  bee  the  face  of  the  cylinder, 
and  c  d  the  face  of  the  front  string.  Makeg-  b  equal  to  %  of 'the 
diameter  of  the  baluster,  and  draw  the  centre-line  of  the  rail,/ g, 
g  hi  and  ij,  parallel  to  a  b,  b  e  c  and  c  d;  make  g  k  and  g  I 
each  equal  to  half  the  width  of  the  rail,  and  through  k  and  /, 
draw  lines  for  the  convex  and  the  concave  sides  of  the  rail,  parallel 
to  the  centre-line ;  tangical  to  the  convex  side  of  the  rail,  and  parallel 
to  k  m,  draw  no;  obtain  the  stretch-out,  q  r,  of  the  semi-circle,  k 
p  m,  according  to  Art.  92  ;  extend  a  b  to  /,  and  k  m  to  s;  make  c  s 
equal  to  the  length  of  the  steps,  and  i  u  equal  to  18  inches,  and  de- 
scribe the  arcs,  s  t  and  u  6,  parallel  to  mp  ;  from  t,  draw  t  w,  tend- 
ing to  the  centre  of  the  cylinder  ;  from  6,  and  on  the  line,  6  ux,  run 
off  the  regular  tread,  as  at  5,  4,  3,  2,  1  and  v  ;  make  u  x  equal  to 
half  the  arc,'w  6,  and  make  the  point  of  division  nearest  to  x.  as 
r,  the  limit  of  the  parallel  steps,  or  flyers  ;  make  r  o  equal  to  m  z  ; 
from  o,  draw  o  a"1,  at  right  angles  to  n  o.  and  equal  to  one  rise; 


353 


AMERICAN    HOUSE-CARPENTEB. 


Fig.  814 

from  a2,  draw  a2  s,  parallel  to  n  o,  and  equal  to  one  tread ;  from  s 
through  0,  draw  s  62. 

Then  from  w,  draw  w  c2,  at  right  angles  to  n  o,  and  set  up,  on 
the  line,  w  c2,  the  same  number  of  risers  that  the  floor,  A,  is  above 
the  first  winder,  J5,  as  at  1,  2,  3,  4,  5  and  6;  through  5,  (on  the 
arc,  6  Uj)  draw  rf2  e2,  tending  to  the  centre  of  the  cylinder ;  from 
c2,  draw  e2/2,  at  right  angles  to  n  o,  and  through  5,  (on  the  line, 


STAIRS.  359 

w  b2,)  draw  g-2/2,  parallel  to  n  o  ;  through  6,  (on  the  line,  w  c2,} 
and/2,  draw  the  line,  h?  b: ;  make  6  c2  equal  to  half  a  rise,  and 
from  ca  and  6,  draw  c2  i2  and  6/,  parallel  to  n  o  ;  make  7i2  r  equal 
to  A2/2;  from  i2,  draw  iU2,  at  right  angles  to  i2  A2,  and  from/-, 
draw/2  4s,  at  right  angles  to/2  A2;  upon  F,  with  A;2/2  for  radius, 
describe  the  arc,/2  i2;  make  62  Z2  equal  to  62/2,  and  ease  off  the 
angle  at  by  by  the  curve,/2  P.  In  the  figure,  the  curve  is  de- 
scribed from  a  centre,  but  in  a  full-size  plan,  this  would  be  imprac- 
ticable ;  the  best  way  to  ease  the  angle,  therefore,  would  be  with 
a  tanged  curve,  according  to  Art.  89.  Then  from  1,2,3  and  4, 
(on  the  line,  w  c2,)  draw  lines  parallel  to  n  o,  meeting  the  curve  in 
w2,  w2,  o2  and  /?2  /  from  these  points,  draw  lines  at  right  angles  to 
n  0,  and  meeting  it  in  a-2,  r2,  s"2  and  f ;  from  x*  and  r2,  draw  lines 
tending  to?*2,  and  meeting  the  convex  side  of  the  rail  in  y2  and 
z* ;  make  m  v2  equal  to  r  s2,  and  m  w*  equal  to  r  f  ;  from  y2,  z\ 
?;2,  and  w2,  through  4,  3,  2  and  1,  draw  lines  meeting  the  line  of 
the  wall-string  in  a3,  63,  c3  and  cP  ;  from  e3,  where  the  centre-line  ot 
the  rail  crosses  the  line  of  the  floor,  draw  e3/3,  at  right  angles  to  n 
o,  and  from  f3,  through  6,  draw/3  g* ;  then  the  heavy  lines,/3 g-2, 
e2  d2,  y2  a3,  z~  b\  v1  c3,  v?  <P,  and  z  y,  will  be  the  lines  for  the  risers, 
which,  being  extended  to  the  line  of  the  front  string,  b  e  c  d,  will 
give  the  dimensions  of  the  winders,  and  the  grading  of  the  front 
string,  as  was  required. 

479. —  To  obtain  the  falling-mould  for  the  twists  of  the  last- 
mentioned  stairs.  Make  ? g3  and  f  h\  (Fig.  314,)  each  equal 
to  half  the  thickness  of  the  rail ;  through  h3  and  g-3,  draw  A3  f 
and  g*f,  parallel  to  r  s  ;  assuming  k  k*  and  m  m?  on  the  plan  as 
the  amount  of  straight  to  be  got  out  with  the  twists,  make  n  q 
equal  to  k  &3,  and  r  /3  equal  to  m  m3 ;  from  n  and  T3,  draw  lines  at 
right  angles  to  n  o,  meeting  the  top  of  the  falling-mould  in  n"  and 
o3;  from  o3,  draw  a  line  crossing  the  falling-mould  at  right  angles 
to  a  chord  of  the  curve,/2  ZJ;  through  the  centre  of  the  cylinder, 
draw  i?  8,  at  right  angles  to  n  o  :  through  8,  draw  7  9,  tending  tc 
A:2;  thenn8  7  will  be  the  falling-mould  for  the  upper  twist,  and  7 
oa  the  falling-mould  for  the  lower  twist. 


360 


AMERICAN    HOUSE-CARPENTER. 


480. —  To  obtain  the  face-moulds.  The  moulds  for  the  twists 
of  this  stairs  may  be  obtained  as  at  Art.  473  ;  but,  as  the  falling- 
mould  in  its  course  departs  considerably  from  a  straight  line,  it 
would,  according  to  that  method,  require  a  very  thick  plank  for 
the  rail,  and  consequently  cause  a  great  waste  of  stuff.  In  order, 
therefore,  to  economize  the  material,  the  following  method  is  to 
be  preferred — in  which  it  will  be  seen  that  the  heights  are  taken 
in  three  places  instead  of  two  only,  as  is  done  in  the  previous 
method. 


Fig.  815.      o 

CASE  1. —  When  the  middle  height  is  above  a  line  joining 
the  other  two.  Having  found  at  Fig.  314  the  direction  of  the 
joint,  w  s3  and  p  e,  according  to  Art.  473,  make  k  pea,  (Fig. 
315,)  equal  to  &3  p3  c  p  in  Fig.  314  ;  join  b  and  c,  and  from  o, 
draw  o  A,  at  right  angles  to  b  c  ;  obtain  the  stretch-out  of  d  g,  as 
df,  and  at  Fig.  314,  place  it  from  the  axis  of  the  cylinder,  p,  to 
q3  ;  from  q3  in  that  figure,  draw  q3  r3,  at  right  angles  to  n  o  ;  also, 
at  a  convenient  height  on  the  line,  n  w3,  in  that  figure,  and  at 
right  angles  to  that  line,  draw  u3  v3 ;  from  b  and  c,  in  Fig.  315, 


STAIRS.  36' 

draw  b  j  and  c  /,  alright  angles  to  b  c  ;  make  bj  equal  to  u*  n3  in 
Fig:  314:,  i  h  equal  to  w3  r3  in  that  figure,  and  c  J  equal  to  v3  9  ; 
from  I,  through  j,  draw  lm  •  from  A,  draw  A  n:  parallel  to  c  6  ; 
from  n.  draw  w  r,  at  right  angles  to  b  c,  and  join  r  and  5  ;  through 
the  lowest  corner  of  the  plan,  as  /?,  draw  v  e,  parallel  to  b  c  ;  from 
a,  e,  «,  p,  &,  £,  and  from  as  many  other  points  as  is  thought  ne- 
cessary, draw  ordinates  to  the  base-line,  v  e,  parallel  to  r  s ; 
through  A,  draw  w  x,  at  right  angles  to  in  1;  upon  n,  with  r  s  for 
radius,  describe  an  intersecting  arc  at  x,  and  join  n  and  x;  from 
the  points  at  which  the  ordinates  from  the  plan  meet  the  base- 
line, v  e,  draw  ordinates  to  meet  the  line,  m  I,  at  right  angles  to  v 
e  ;  and  from  the  points  of  intersection  on  m  /,  draw  correspond- 
ing ordinates,  parallel  to  nx  ;  make  the  ordinates  which  are  pa- 
rallel to  n  x  of  a  length  corresponding  to  those  which  are  parallel 
to  r  5,  and  through  the  points  thus  found,  trace  the  face-mould 
as  required. 

CASE  2. —  When  the  middle  height  is  below  a  line  joining 
the  other  two.  The  lower  twist  in  Fig.  314  is  of  this  nature. 
The  face-mould  for  this  is  found  at  Fig.  316  in  a  manner  similar 
to  that  at  Fig.  315.  The  heights  are  all  taken  from  the  top  of 
the  falling-mould  at  Fig.  314 ;  bj  being  equal  to  M?  6  in  Fig.  314, 
i  h  equal  to  x5  y3  in  that  figure,  and  c  I  to  I3  o3.  Draw  a  line 
th rough  J  and  Z,  and  from  h,  draw  h  n,  parallel  to  b  c  ;  from  w, 
draw  n  r,  at  right  angles  to  b  c,  and  join  r  and  s  ;  then  r  s  will  be 
the  bevil  for  the  lower  ordinates.  From  h,  draw  h  x,  at  right  an- 
gles to  j  I  /  upon  n,  with  r  s  for  radius,  describe  an  intersecting 
arc  at  x,  and  join  n  and  x  ;  then  n  x  will  be  the  bevil  for  the  upper 
ordinates,  upon  which  the  face-mould  is  found  as  in  Case  1. 

481. — Elucidation  of  the  foregoing  method. — This  method 
of  finding  the  face-moulds  for  the  handrailing  of  winding  stairs, 
being  founded  on  principles  which  govern  cylindric  sections,  may 
be  illustrated  by  the  following  figures.  Fig.  317  and  318  repre- 
sent solid  blocks,  or  prisms,  standing  upright  on  a  level  base,  b  d  ; 
the  upner  surface,  j  a  forming  oblique  angles  with  the  face,  6  / — 
46 


AMERICAN    HOUSE-CARPENTER. 


Fig.  816. 


in  Fig.  317  obtuse,  and  in  Fig.  318  acute.  Upon  the  base,  de 
scribe  the  semi-circle,  b  s  c  ;  from  the  centre,  *,  draw  i  .?,  at  right 
angles  to  b  c  ;  from  s,  draw  5  x,  at  right  angles  to  e  d,  and  from  i 
draw  i  A,  at  right  angles  to  b  c  ;  make  i  h  equal  to  5  x.  and  join 
h  and  r  ;  then,  h  and  #  being  of  the  same  height,  the  line,  h  .r. 
joining  them,  is  a  level  line.  From  h,  draw  h  n,  parallel  to  b  c, 
and  from  ?i:  draw  n  r,  at  right  angles  to  b  c ;  join  r  and  s,  also  w 


&TAIRS. 


3G3 


Fig.  81T. 


Fig.  SIS. 


and  x ••;  then,  n  and  x  being  of  the  same  height,  n  x  is  a  'evel  line ; 
and  this  line  lying  perpendicularly  over  r  s,  n  x  and  r  s  must  be 
of  the  same  length.  So,  all  lines  on  the  top,  drawn  parallel  to  n 
x,  and  perpendicularly  over  corresponding  lines  drawn'parallel  to 
r  s  on  the  base,  must  be  equal  to  those  lines  on  the  base ;  and  by 
drawing  a  number  of  these  on  the  semi-circle  at  the  base  and 
others  of  the  same  length  at  the  top,  it  is  evident  that  a  curve,  j 
x  /,  may  be  traced  through  the  ends  of  those  on  the  top,  which 
shall  lie  perpendicularly  over  the  semi-circle  at  the  base. 

It  is  upon  this  principle  that  the  process  at  Fig.  315  and  316 
is  founded.  The  plan  of  the  rail  at  the  bottom  of  those  figures 
is  supposed  to  lie  perpendicularly  under  the  face-mould  at  the  top ; 
and  each  ordinate  at  the  top  over  a  corresponding  one  at  the  base. 
The  ordinates,  n  x  and  r  s,  in  those  figures,  correspond  to  n  'x 
and  r  sin  Fig.  317  and  318. 

In  Fig.  319,  the  top,  e  a,  forms  a  right  angle  with  the  face,  d 
c  ;  all  that  is  necessary,  therefore,  in  this  figure,  is  to  find  a  line 
corresponding  to  h  x  in  the  last  two  figures,  and  that  will  lie  level 
and  in  the  upper  surface ;  so  that  all  ordinates  at  right  angles  to 
d  r  on  the  base,  will  correspond  to  those  that  are  at  right  angles 


364 


AMERICAN    HOUSE  CARPENTEH. 


Fig  319  r 


to  c  c  on  the  top.     This  elucidates  Fig.  307 ;  at  wjiich  the  lines, 
n  9  and  i  8.  correspond  to  h  9  and  i  S  in  this  figuie. 


Fig.  320. 


482.—  To  find  the  bevil  for' the  edge  of  the  plank.  The 
plank,  before  the  face-mould  is  applied,  must  be  bevilled  accord- 
ing to  the  angle  which  the  top  of  the  imaginary  block,  or  prism, 
in  the  previous  figures,  makes  with  the  face.  This  angle  is  de- 
termined in  the  following  manner  :  draw  w  i,  (Fig.  320,)  at  right 
angles  to  i  s,  and  equal  to  w  h  at  Fig.  SI  5 ;  make  i  s  equal  to  i  s  in 
that  figure,  and  join  w  and  s  ;  then  5  w  p  will  be  the  bevil  required 
in  order  to  apply  the  face-mould  nt  Fig.  315.  In  Fig.  316,  the 
middle  height  being  below  the  line  joining  the  other  two,  the  bevil 
is  therefore  acute.  To  determine  this,  draw  i  s,  (Fig.  321,)  at 


305 


Fig.  821. 


right  angles  to  i  p,  at  d  equal  to  i  s  in  Fig.  316  ;  make  s  t>  equal 
to  h  w  in  Fig:  316,  and  join  w  and  i  ;  then  w  i  p  will  be  the 
bevil  required  in  ordsr  to  apply  the  face-mould  at  Fig.  316.  Al 
though  the  falling-mould  in  these  cases  is  curved,  yet,  as  the 
plank  is  sprung,  or  bevilled  on  its  edge,  the  thickness  necessary 
to  get  out  the  twist  may  be  ascertained  according  to  Art.  474 — 
taking  the  vertical  distance  across  the  falling-mould  at  the  joints, 
and  placing  it  down  from  the  two  outside  heights  in  Fig.  315  or 
316.  After  bevilling  the  plank,  the  moulds  are  applied  as  at  Art. 
476 — applying  the  pitch-board  on  the  bevilled  instead  of  a  square 
edge,  and  placing  the  tips  of  the  mould  so  that  they  will  bear  the 
same  relation  to  the  edge  of  the  plank,  as  they  do  to  the  line,  j  I, 
in  Fig.  315  or  316. 


Fig.  322. 


483. To   apply  the  moulds  without  bevilling  the  plank. 

Make  w  p,  (Fig.  322,)  equal  to  w  p  at  Fig.  320,  and  the  angle, 
bed,  equal  to  b  j  I  in  Fig.  315  ;  make  p  a  equal  to  the  thick- 
ness of  the  plank,  as  w  a  ir.  Fig.  320,  and  from  a  draw  a  o.  pa- 
rallel to  w  d  ;  from  c,  draw  c  e,  at  right  angles  to  w  d,  and  join  e 


366 


AMERICAN    HOUSE-CARPENTER, 


and  b  ;  then  the  angle,  b  e  o}  on  a  square  edge  of  the  plank,  hav 
ing  a  line  on  the  upper  face  at  the  distance,  p  a,  in  Fig.  320,  al 
which  to  apply  the  tips  of  the  mould — will  answer  the  same  pur- 
pose as  bevilling  the  edge. 

If  the  bevilled  edge  of  the  plank,  which  reaches  from  p  to  w, 
is  supposed  to  be  in  the  plane  of  the  paper,  and  the  point,  a,  to 
be  above  the  plane  of  the  paper  as  much  as  a,  in  Fig.  320,  is  dis- 
tant from  the  line,  w  p  ;  and  the  plank  to  be  revolved  on  p  b  as 
an  axis  until  the  line,  p  w,  falls  below  the  plane  of  the  paper,  and 
the  line,  p  a,  arrives  in  it ;  then,  it  is  evident  that  the  point,  c} 
will  fall,  in  the  line,  c  e,  until  it  lies  directly  behind  the  point,  e, 
and  the  line,  b  c,  will  lie  directly  behind  b  e. 


Fig.  823. 


484. —  To  find  the  bevils  for  splayed  work.  The  principle 
employed  in  the  last  figure  is  one  that  will  serve  to  find  the  bevils 
for  splayed  work — such  as  hoppers,  bread-trays,  &c. — and  a  way 
of  applying  it  to  that  purpose  had  better,  perhaps,  be  introduced 
in  this  connection.  In  Fig.  323,  a  b  c  is  the  angle  at  which  the 
work  is  splayed,  and  6  d,  on  the  upper  edge  of  the  board,  is  at 
right  angles  to  a  b  ;  make  the  angle,  fgj,  equal  to  a  b  c,  and 
from/,  draw  f  h,  parallel  to  e  a;  from  6,  draw  b  o,  at  right  an 
gks  to  a  b  ;  through  o,  draw  i  e,  parallel  to  c  6,  and  join  e  and 
d  ;  then  the  angle,  a  e  d,  will  be  the  proper  bevil  for  the  ends  from 
the  inside,  or  k  d  e  from  the  outside.  If  a  mitre-joint  is  ie- 


STAIRS.  367 

quired,  setfg,  the  thickness  of  the  stuff  on  the  level,  from  e  to 
m,  and  join  m  and  d  ;  then  A*  d  m  will  be  the  proper  bevil  for  a 
mitre-joint. 

If  the  upper  edges  of  the  splayed  work  is  to  be  bevilled,  so  as 
to  be  horizontal  when  the  work  is  placed  in  its  proper  position, 
/£*  j)  being  the  same  as  a  b  c,  will  be  the  proper  bevil  for  that 
purpose.  Suppose,  therefore,  that  a  piece  indicated  by  the  lines, 
kg,  gf  and  f  h,  were  taken  off;  then  a  line  drawn  upon  the 
bevilled  surface  from  d,  at  right  angles  to  k  d,  would  show  the 
true  position  of  the  joint,  because  it  would  be  in  the  direction  of 
the  board  for  the  other  side ;  but  a  line  so  drawn  would  pass 
through  the  point,  o, — thus  proving  the  principle  correct.  So,  if 
a  line  were  drawn  upon  the  bevilled  surface  from  d,  at  an  angle 
of  45  degrees  to  A*  d,  it  would  pass  through  the  point,  n. 

485. — Another  method  for  face-moulds.  It  will  be  seen  by 
reference  to  Art.  481,  that  the  principal  object  had  in  view  in  the 
preparatory  process  of  finding  a  face-mould,  is  to  ascertain  upon  it 
the  direction  of  a  horizontal  line.  This  can  be  found  by  a  method 
different  fro'm  any  previously  proposed  ;  and  as  it  requires  fewer 
lines,  and  admits  of  less  complication,  it  is  probably  to  be  preferred. 
It  can  be  best  introduced,  perhaps,  by  the  following  explanation . 

In  Fig.  324,  j  d  represents  a  prism  standing  upon  a  level  base, 
b  d.  its  upper  surface  forming  an  acute  angle  with  the  face, 
b  I,  as  at  Fig.  318.  Extend  the  base  line,  b  c,  and  the  raking 
line,  j  /,  to  meet  at/;  also,  extend  e  d  and  g  a,  to  meet  at  Ar; 
from  /,  through  A',  draw  /  m.  If  we  suppose  the  prism  to  stand 
upon  a  level  floor,  ofm,  and  the  plane,  j  g  a  I,  to  be  extended 
to  meet  that  floor,  then  it  -will  be  obvious  that  the  intersection 
Detween  that  plane  and  the  plane  of  the  floor  would  be  in  the  line, 
fk;  and  the  line,/ k,  being  in  the  plane  of  the  floor,  and  also  in 
the  inclined  plane,  j  g  kf,  any  line  made  in  the  plane,  j  g  kf. 
parallel  to/  A-,  must  be  a  level  line.  By  finding  the  position  of  a 
perpendicular  plane,  at  right  angles  to  the  raking  plane,  j/  k  g} 
we  shall  greatly  shorten  the  process  for  obtaining  ordinates. 


368 


AMERICAN    HOUSE-CARPENTER. 


Fig.  824  / 


This  may  be  done  thus  :  from/,  draw/o.  at  right  angles  tofm; 
extend  e  b  to  o,  andg-j,  to  t ;  from  o,  draw  o  /,  at  right  angles  to 
o/,  and  join  t  and/;  then  £  o/  will  be  a  perpendicular  plane,  at 
right  angles  to  the  inclined  plane,  t  g  kf;  because  the  base  of 
the  former,  o/  is  at  right  angles  to  the  base  of  the  latter,/ A',  both 
these  lines  being  in  the  same  plane.  From  6,  draw  b  p,  at  right 
angles  to  o/  or  parallel  tofm  ;  from  p.  draw  p  q,  at  right  angles 
to  of,  and  from  q,  draw  a  line  on  the  upper  plane,  parallel  tofm, 
or  at  right  angles  to  tf;  then  this  line  will  obviously  be  drawn 
to  the  point,  j,  and  the  line,  q  j,  be  equal  top  b.  Proceed,  in  the 
same  way,  from  the  points,  s  and  c,  to  find  x  and  I. 

Now,  to  apply  the  principle  here  explained,  let  the  curve,  b  s  c, 
(Fig.  325,)  be  the  base  of  a  cylindric  segment,  and  let  it  be  re- 
quired to  find  the  shape  of  a  section  of  this  segment,  cut  by  a 
plane  passing  through  three  given  points  in  its  curved  surface : 
one  perpendicularly  over  6,  at  then-eight,  bj;  one  perpendicu- 
larly over  s,  at  the  height,  s  x  ;  and  the  other  over  c,  at  the  height, 
c  I — these  lines  being  drawn  at  right  angles  to  the  chord  of  the 
base,  b  c.  From^',  through  /,  draw  a  line  to  meet  the  chord  line 
extended  to//  from  s,  draw  s  k,  parallel  to  b  /,  and  from  #, 
draw  x  k,  parallel  tojf;  from/,  through  k,  draw//w;  then/wi 
will  be  the  intersecting  line  of  »he  plane  of  the  section  with  ths 


309 


plane  of  the  base.  This  line  can  be  proved  to  be  the  intersection 
of  these  planes  in  another  way  ;  from  6,  through  s,  and  from  j, 
through  .r,  draw  lines  meeting  at  m  ;  then  the  point,  m,  will  be 
in  the  intersecting  line,  as  is  shown  in  the  figure,  and  also  at 
Fig.  324. 

From/,  draw/p,  at  right  angles  tofm;  from  b  and  c,  and 
from  as  many  other  points  as  is  thought  necessary,  draw  ordinates, 
parallel  tofm;  make  p  q  equal  to  bj,  and  join  q  and//  from 
the  points  at  which  the  ordinates  meet  the  line,  qf,  draw  others 
at  right  angles  to  q  f;  make  each  ordinate  at  A  equal  to  its  cor- 
responding ordinate  at  C,  and  trace  the  curve,  gn  i,  through  the 
points  thus  found. 

Now  it  may  be  observed  that  A  is  the  plane  of  the  section,  B 
the  plane  of  the  segment,  corresponding  to  the  plane,  q  p  f,  of 
Fig.  324,  and  C  is  the  plane  of  the  base.  To  give  these  planes 
their  projer  position,  let  A  be  turned  on  q  f  as  an  axis  until  it 


270  AMERICAN    HOUSE-CARPENTER. 

stands  perpendicularly  over  the  line,  qf,  and  at  right  angles  tc 
the  plane,  B  ;  then,  while  A  and  B  are  fixed  at  right  angles,  let 
B  be  turned  on  the  line,  p  /,  as  an  axis  until  it  stands  perpendicu- 
larly over  pf,  and  at  right  angles  to  the  plane,  C  ;  then  the  plane, 
A,  will  lie  over  the  plane,  C,  with  the  several  lines  on  one  corres- 
ponding to  those  on  the  other  ;  the  point,  t,  resting  at  I,  the  point, 
n,  at  a:,  and  g  at  j  ;  and  the  curve,  g  n  i:  lying  perpendicularly 
over  b  s  c — as  was  required.  If  we  suppose  the  cylinder  to  be 
cut  by  a  level  plane  passing  through  the  point,  Z,  (as  is  done  in 
finding  a  face-mould,)  it  will  be  obvious  that  lines  corresponding 
to  q  f  and  pf  would  meet  in  /;  and  the  plane  of  the  section,  A. 
the  plane  of  the  segment,  B,  and  the  plane  of  the  base,  C,  would 
all  meet  in  that  point. 

486. —  To  find  the  face- mould  for  a  hand-rail  according  to 
the  principles  explained  in  the  previous  article.  In  Fig.  326, 
a  e  cf  is  the  plan  of  a  hand-rail  over  a  quarter  of  a  cylinder ;  and 
in  Fig.  327,  a  b  c  d  is  the  falling-mould ;  /  e  being  equal  to  the 
stretch-out  of  a  df  in  Fig.  326.  From  c,  draw  c  /i,  parallel  to 
e  f ;  bisect  h  c  in  i,  and  find  a  point,  as  b,  in  the  arc,  df,  (Fig. 
326.)  corresponding  to  i  in  the  line,  h  c;  from  i,  (Fig.  327,)  to 
the  top  of  the  falling-mould,  draw  i  j, at  right  angles iohc;  at  Fig. 
326,  from  c,  through  b,  draw  c  g,  and  from  b  and  c,  draw  b  j  and 
c  /«;,  at  right  angles  to  g  c  ;  make  c  k  equal  to  h  g  at  Fig.  327, 
and  b  j  equal  to  ij  at  that  figure  ;  from  k,  through  j,  draw  k  g, 
and  fromg-,  through  a,  draw  g  p  ;  then  gp  will  be  the  intersecting 
line,  corresponding  \ofm  in  Fig.  324  and  325 ;  through  e,  draw 
p  6,  at  right  angles  to  gp,  arid  from  c,  draw  c  q.  parallel  to  g  p  ; 
make  r  q  equal  to  h  g  at  Fig.  327  ;  join/>  and  </,  and  proceed  as 
in  the  previous  examples  to  find  the  face-mould,  A.  The  joint 
of  the  face-mould,  u  v,  will  be  more  accurately  determined  by 
finding  the  projection  of  the  centre  of  the  plan,  o,  as  at  w ; 
joining  s  and  w,  and  drawing  u  v,  parallel  to  s  w. 

It  may  be  noticed  that  c  k  and  b  j  are  not  of  a  length  corres- 
ponding to  the  above  directions  :  they  are  but  %  the  length  given, 


372 


AMERICAN    HOUSE  rARPENTER. 


V. 


The  object  of  drawing  these  lines  is  to  find  the  point,  g,  and  that 
can  be  done  by  taking  any  proportional  parts  of  the  lines  given, 
as  well  as  by  taking  the  whole  lines.  For  instance,  supposing  c 
k  and  b  j  to  be  the  full  length  of  the  given  lines,  bisect  one  in  i 
and  the  other  in  m;  then  a  line  drawn  from  ra,  through  «,  will 
give  the  point,  g,  as  was  required.  The  point,  g,  may  also  be 


STAIRS.  373 

obtained  thus :  at  Fig.  327,  make  h  I  equal  to  c  b  in  Fig.  326 
irom  /,  draw  I  k,  at  right  angles  to  h  c  ;  from  j,  draw^*  k,  parallel 
to  h  c  ;  from  g,  through  k,  draw  g  n  ;  at  Fig.  326,  make  b  g 
equal  to  I  n  in  Fig.  327 ;  then  g  will  be  the  point  required. 

The  reason  why  the  points,  a,  b  and  c,  in  the  plan  of  the  rail  at 
Fig.  326,  are  taken  for  resting  points  instead  of  e,  i  and/,  is  this  : 
the  top  of  the  rail  being  level,  it  is  evident  that  the  points,  a  and  e, 
in  the  section  a  e,  are  of  the  same  height ;  also  that  the  point,  i,  is  ot 
the  same  height  as  6,  and  c  as/.  Now,  if  a  is  taken  for  a  point 
in  the  inclined  plane  rising  from  the  line  g  p,  e  must  be  below 
that  plane  ;  if  b  is  taken  for  a  point  in  that  plane,  i  must  be  below 
it ;  and  if  c  is  in  the  plane,/  must  be  below  it.  The  rule,  then, 
for  taking  these  points,  is  to  take  in  each  section  the  one  that  is 
nearest  to  the  line,  g  p.  Sometimes  the  line  of  intersection,  g  p, 
happens  to  come  almost  in  the  direction  of  the  line,  e  r  :  in  such 
case,  after  finding  the  line,  see  if  the  points  from  which  the 
heights  were  taken  agree  with  the  above  rule ;  if  the  heights 
were  taken  at  the  wrong  points,  take  them  according  to  the  rule 
above,  and  then  find  the  true  line  of  intersection,  which  will  not 
vary  much  from  the  one  already  found. 


487._Tb  apply  the  face-mould  thus  found  to  the  plank. 
The  face-mould,  when  obtained  by  this  method,  is  to  be  applied 
to  a  square-edged  plank,  as  directed  at  Art.  476,  with  this  differ- 
ence :  instead  of  applying  both  tips  of  the  mould  to  the  edge  of 


374:  AMERICAN    HOUSE-CARPENTER. 

the  plank,  one  of  them  is  to  be  set  as  far  from  the  edge  of  the 
plank,  as  #,  in  Fig.  326,  is  from  the  chord  of  the  section  p  q — as 
is  shown  at  Fig.  328.  A,  in  this  figure,  is  the  mould  applied  on 
the  upper  side  of  the  plank,  B:  the  edge  of  the  plank,  and  C,  the 
mould  applied  on  the  under  side  ;  a  b  and  c  d  being  made  equal 
to  q  x  in  Fig.  326,  and  the  angle,  e  a  c,  on  the  edge,  equal  to  the 
angle,  p  q  r,  at  Fig.  326,  In  order  to  avoid  a  waste  of  stuff,  it 
would  be  advisable  to  apply  the  tips  of  the  mould,  e  and  b,  im- 
mediately at  the  edge  of  the  plank.  To  do  this,  suppose  the 
moulds  to  be  applied  as  shown  in  the  figure ;  then  let  A  be  re- 
volved upon  e  until  the  point,  6,  arrives  at  g,  causing  the  line,  e  b, 
to  coincide  with  eg:  the  mould  upon  the  under  side  of  the 
plank  must  now  be  revolved  upon  a  point  that  is  perpendicularly 
beneath  e,  as  /;  from/,  draw  /  A,  parallel  to  i  d,  and  from  d} 
draw  d  h,  at  right  angles  to  i  d  ;  then  revolve  the  mould,  C,  upon 
/,  until  the  point,  h,  arrives  at  j,  causing  the  line,/  A,  to  coincide 
wilhfj,  and  the  line,  i  d,  to  coincide  with  k  I  •  then  the  tips  of 
the  mould  will  be  at  k  and  /. 

The  rule  for  doing  this,  then,  will  be  as  follows :  make  the  an- 
gle, ifk,  equal  to  the  angle  q  v  x,  at  Fig.  326  ;  make  fk  equal 
to/i,  and  through  A:,  draw  A;  Z,  parallel  to  ij ;  then  apply  the 
corner  of  the  mould,  i,  at  k,  and  the  other  corner  d,  at  the  line,  k  I. 
The  thickness  of  stuff  is  found  as  at  Art.  474. 
488. —  To  regulate  the  application  of  the  falling-mould. 
Obtain,  on  the  line,  h  c,  (Fig.  327,)  the  several  points,  r,  q,p,  I 
and  m,  corresponding  to  the  points,  6',  a3,  z,  y,  &c.,  at  Fig.  326  , 
from  r  q  p,  &c.,  draw  the  lines,  r  t,  q  u,p  v,  &c.,  at  right  angles 
to  h  c;  make  h  s,  r  t,  q  «,  &c.,  respectively  equal  to  6  c2,  r  y,  5 
d\  &c.,  at  Fig.  326 ;  through  the  points  thus  found,  trace  the 
curve,  s  w  c.  Then  get  out  the  piece,  g  s  c,  attached  to  the  fall- 
ing-mould at  several  places  alor.g  its  length,  as  at  z:  z,  z,  &c. 
In  applying  the  falling-mould  with  this  strip  thus  attached,  the 
edge,  sw  c,  will  coincide  with  the  upper  surface  of  the  rail  piece 


STAIRS.  375 

before  it  is  squared  ;  and  thus  show  the  proper  position  ->f  the  fall- 
ing-mould along  its  whole  length.   (See  Art,  496.) 


SCROLLS    FOR    HAND-RAILS. 

*s9. —  General  rule  for  finding  the  size  and  position  of  tka 
regulating  square.  The  breadth  which  the  scroll  is  to  occupy, 
the  number  of  its  revolutions,  and  the  relative  size  of  the  reguia 
ting  square  to  the  eye  of  the  scroll,  being  given,  multiply  the 
number  of  revolutions  by  4,  and  to  the  product  add  the  number 
of  times  a  side  of  the  square*  is  contained  in  the  diameter  of  the 
eye,  and  the  sum  Avill  be  the  number  of  equal  parts  into  which 
the  breadth  is  to  be  divided.  Make  a  side  of  the  regulating 
square  equal  to  one  of  these  parts.  To  the  breadth  of  the  scroll 
add  one  of  the  parts  thus  found,  and  half  the  sum  will  be  the 
length  of  the  longest  ordinate. 


Fig.  829 


490. —  To  find  the  proper  centres  in  the  regulating  square. 
Let  a  2  1  b,  (Fig.  329,)  be  the  size  of  a  regulating  square,  found 
according  to  the  previous  rule,  the  required  number  of  revolu- 
tions being  If.  Divide  two  adjacent  sides,  as  a  2  and  2  1,  into 
as  many  equal  parts  as  there  are  quarters  in  the  number  of  revo- 
lutions, as  seven  ;  from  those  points  of  division,  draw  lines  across 
the  square,  at  right  angles  to  the  lines  divided ;  then,  1  being  the 
first  centre,  2,  3,  4,  5,  6  and  7,  are  the  centres  for  the  other  quar 
ters,  and  8  is  the  centre  for  the  eye  ;  the  heavy  lines  that  deter- 


376 


AMERICAN    HOUSE-CARPENTER 


mine  these  centres  being  each  one  part  less  in  length  than  its  pre 
ceding  line. 


Fig.  830. 


491. —  To  describe  the  scroll  for  a  hand-rail  over  a  curtail 
step.  Let  a  b,  (Fig.  330,)  be  the  given  breadth,  If  the  given 
number  of  revolutions,  and  let  the  relative  size  of  the  regulating 
square  to  the  eye  be  £  of  the  diameter  of  the  eye.  Then,  by  the 
rale,  If  multiplied  by  4  gives  7,  and  3,  the  number  of  times  a 
side  of  the  square  is  contained  in  the  eye,  being  added,  the  sum 
is  10.  Divide  a  6,  therefore,  into  10  equal  parts,  and  set  one  from 
b  to  c  ;  bisect  a  c  in  e  ;  then  a  c  will  be  the  length  of  the  longest 
ordinate,  (1  d  or  1  e.)  From  a.  draw  a  d,  from  e,  draw  e  1,  and 
from  6,  draw  &/,  all  at  right  angles  to  a  b  ;  make  e  1  equal  to  e 
a,  and  through  1,  draw  1  rf,  parallel  to  a  b  ;  set  b  c  from  1  to  2, 
and  upon  1  2.  complete  the  regulating  square  ;  divide  this  square 
as  at  Fig.  329  ;  then  describe  the  arcs  that  compose  the  scroll,  as 
follows:  upon  1,  describe  de;  upon  2,  describe  ef;  upon  3, 
describe  f  §•  •  upon  4,  describe  g  h,  &c. ;  make  d  /equal  to  the 


STAIRS.  377 

width  of  the  rail,  and  upon  1,  describe  I  m  ;  upon  2,  describe  m 
n,  &c. ;  describe  the  eye  upon  8,  and  the  scroll  is  completed. 

492. —  To  describe  the  scroll  for  a  curtail  step.  Bisect  d  I, 
(Fig:  330,)  in  o,  and  make  o  v  equal  to  J  of  the  diameter  of  a 
baluster ;  make  v  w  equal  to  the  projection  of  the  nosing,  and  e 
x  equal  to  w  1;  upon  1,  describe  w  y,  and  upon  2,  describe  y  z  ; 
also  upon  2,  describes  i;  upon  3,  describe  ij,  and  so  around  to 
z  ;  and  the  scroll  for  the  step  will  be  completed. 

493. —  To  determine  the  position  of  the  balusters  under  the 
scroll  Bisect  d  I,  (Fig.  330,)  in  o,  and  upon  1,  with  1  o  for  ra- 
dius, describe  the  circle,  or  u;  set  the  baluster  at  p  fair  with  the 
face  of  the  second  riser,  c7,  and  from  p,  with  half  the  tread  in  the 
dividers,  space  off  as  at  o,  q,  r,  s,  t,  u,  &c.,  as  far  as  <f ;  upon  2, 
3,  4  and  5,  describe  the  centre-line  of  the  rail  around  to  the  eye 
of  the  scroll ;  from  the  points  of  division  in  the  circle,  o  r  it,  draw 
lines  to  the  centre-line  of  the  rail,  tending  to  the  centre  of  the 
eye,  8 ;  then,  the  intersection  of  these  radiating  lines  with  the 
centre-line  of  the  rail,  will  determine  the  position  of  the  balusters, 
as  shown  in  the  figure. 


4:94:.—  To  obtain  the  falling-mould  for  the  raking  part  of  the 
scroll.  Tangical  to  the  rail  at  h,  (Fig.  330,)  draw  h  k,  parallel  to  d 
a;  then  k  d1  will  be  the  joint  bet  ween  the  twist  and  the  other  part 
of  the  scroll.  Make  d  e*  equal  to  the  stretch-out  of  de,  and  upon  d 

48 


378 


AMERICAN    HOUSE-CARPENTER. 


e2,  find  the  position  of  the  point,  fr,  as  at  A* ;  at  Fig.  331,  make  e  d 
equal  to  e2  d  in  Fig.  330,  and  c?  c  equal  to  tZ  c2  in  that  figure  : 
from  c,  draw  c  a,  at  right  angles  to  e  c,  and  equal  to  one  rise  ; 
make  c  b  equal  to  one  tread,  and  from  6,  through  a,  draw  b  j , 
bisect  a  cinl,  and  through  Z,  draw  m  q,  parallel  to  e  h  ;  m  q  is 
the  height  of  the  level  part  of  a  scroll,  which  should  always  be 
about  3i  feet  from  the  floor ;  ease  oif  the  angle,  m  /j,  according 
to  Art.  89,  and  draw  g  w  n,  parallel  to  m  xj:  and  at  a  distance 
equal  to  the  thickness  of  the  rail ;  at  a  convenient  place  for  the 
joint,  as  i,  draw  i  «,  at  right  angles  to  b  j  ;  through  n,  draw  ;  A, 
at  right  angles  to  e  h  ;  make  d  k  equal  to  d  k*  in  Fig.  330,  and 
from  k)  draw  k  0,  at  right  angles  to  e  h  ;  at  Fig.  330,  make  d 
/i'2  equal  to  d  h  in  Fig.  331,  and  draw  /i2  62,  at  right  angles  to  d 
A2 ;  then  k  a?  and  /i2  i2  will  be  the  position  of  the  joints  on  the 
plan,  and  at  Fig.  331,  o  p  and  i  n,  their  position  on  the  falling- 
mould  ;  and  p  o  i  n,  (Fig.  331,)  will  be  the  falling-mould  re 
quired. 


Fig.  832. 


495.—  To  describe  the  face-mould.  At  Fig.  330,  from  Ar,  dra\\ 
k  r2,  at  right  angles  to  r2  d  ;  at  Fig.  331,  make  h  r  equal  to  h?  r1 
in  Fig.  330,  and  from  r,  draw  r  s,  at  right  angles  to  r  h  ;  from 
the  intersection  of  r  s  with  the  level  line,  m  5-,  through  i,  draw  s 
t ;  at  Fig.  330,  make  A*  62  equal  to  7  £  in  Fig.  331,  and  join  6' 
and  r2 ;  from  a2,  and  from  as  many  other  points  in  the  arcs,  a2  / 
and  k  d,  as  is  thought  necessary,  draw  ordinates  to  r2  d,  at  right 
angles  to  the  latter ;  make  r  b,  (Fig.  332,)  equal  in  its  length  and 
in  its  divisions  to  the  line,'  r2  62,  in  Fig.  330  ;  from  r,  n.  o,  p,  q 


STAIRS. 


379 


and  I,  draw  the  lines,  r  k,  n  d,  o  a,  p  e,  qf  and  /  c,  at  right  an- 
gles to  r  b,  and  equal  to  r*  k,  d?  s\  f  a\  &c.,  in  Fig.  330 ; 
through  the  points  thus  found,  trace  the  curves,  k  I  and  a  c,  and 
complete  the  face-mould,  as  shown  in  the  figure.  This  mould  is 
to  be  applied  to  a  square-edged  plank,  with  the  edge,  I  &,  parallel 
to  the  edge  of  the  plank.  The  rake  lines  upon  the  edge  of  the 
plank  are  to  be  made  to  correspond  to  the  angle,  s  t  A,  in  Figr. 
331.  The  thickness  of  stuff  required  for  this  mould  is  shown  at 
Fig.  331,  between  the  lines  s  t  and  u  v — u  v  being  drawn  pa- 
rallel to  s  t. 

496. — All  the  previous  examples  given  for  finding  face-moulds 
over  winders,  are  intended  for  moulded  rails.  For  round  rails, 
the  same  process  is  to  be  followed  with  this  difference :  instead 
of  working  from  the  sides  of  the  rail,  work  from  a  centre-line. 
After  finding  the  projection  of  that  line  upon  the  upper  plane, 
describe  circles  upon  it,  as  at  Fig.  293,  and  trace  the  sides  of  the 
moulds  by  the  points  so  found.  The  thickness  of  stuff  for  the 
twists  of  a  round  rail,  is  the  same  as  for  the  straight ;  and  the 
twists  are  to  be  sawed  square  through. 

h  /   *        * 


880  AMERICAN    HOUSE-CARPENTER 

497. — To  ascertain  the  form  of  the  newel-cap  from  a  section 
of  the  rail.  Draw- a  b,  (Fig.  333,)  through  the  widest  part  of 
tke  given  section,  and  parallel  to  c  d ;  bisect  a  b  in  e,  and  through 
a,  e  and  &,  draw  h  i,fg,  and  k  j,  at  right  angles  to  a  6  ;  at  a  con- 
venient place  on  the  line,  fg,  as  o,  with  a  radius  equal  to  half 
the  width  of  the  cap,  describe  the  circle,  i  j  g ;  make  r  I  equal 
to  e  b  or  e  a ;  join  /  and  j,  also  I  and  i ;  from  the  curve,  f  b,  to 
the  line,  I  j,  draw  as  many  ordinates  as  is  thought  necessary, 
parallel  to  f  g ;  from  the  points  at  which  these  ordinates  meet 
the  line,  Ij,  and  upon  the  centre,  o,  describe  arcs  in  continuation  to 
meet  o  p  ;  from  n,  t,  x,  &c.,  draw  n  s,  t  u,  &c.,  parallel  to  /  g  ; 
make  n  s,  t  u,  &c.,  equal  to  e  f,  w  v,  &c. ;  make  x  y,  &c.,  equal 
to  z  d,  &c. ;  make  o  2,  o  3,  &c.,  equal  to  o  n,  o  t,  &c. ;  make  2  4 
equal  to  n  s,  and  in  this  way  find  the  length  of  the  lines  crossing 
o  m ;  through  the  points  thus  found,  describe  the  section  of  the 
newel-cap,  as  shown  in  the  figure. 


Fig    334. 

498. — To  find  the  true  position  of  a  butt  joint  for  the  twists  of 
a  moulded  rail  over  platform  stairs.  Obtain  the  shape  of  the 
mould  according  to  Art.  466,  and  make  the  line  a  b,  Fig.  334, 
equal  to  a  c,  Fig.  300  ;  from  6,  draw  b  c,  at  right  angles  to  a  b, 
and  equal  in  length  to  n  m,  Fig.  300  :  join  a  and  c,  and  bisect  a  c 
in  o ;  through  o  draw  e  f,  at  right  angles  to  a  c,  and  d  k,  parallel 
to  c  b  •  make  o  d  and  o  k  each  equal  to  half  e  h  at  Fig.  300 ; 
through  e  and  /,  draw  h  i  and  gj,  parallel  to  o  c.  At  Fig.  301, 
make  n  a  equal  to  e  d,  Fig.  334,  and  through  a,  draw  r  p,  at  right 
angles  to  n  c  ;  then  r  p  will  be  the  true  position  on  the  face-mould 
for  a  butt  joint,  as  was  required.  The  sides  must  be  sawn  verti 
i 


STAIRS. 


381 


cally  as  described  at  Art.  467,  but  the  joint  is  to  be  sawn  square 
through  the  plank.  The  moulds  obtained  for  round  rails,  (Art. 
464,)  give  the  line  for  the  joint,  when  apj.lied  to  either  side  of  the 
plank ;  but  here,  for  moulded  rails,  tho  line  for  the  joint  can  be 
obtained  from  only  one  side.  Whe  i  the  rail  is  canted  up,  the 
joint  is  taken  from  the  mould  laid  on  the  upper  side  of  the  lowei 
twist,  and  on  the  under  side  of  the  upper  twist ;  but  when  it  is 
canted  down,  a  course  just  the  reverse  of  this  is  to  be  pursued. 
When  the  rail  is  not  canted,  either  up  or  down,  the  vertical  joint, 
obtained  as  at  Art.  466,  will  be  a  butt  joint,  and  therefore,  in  such 
a  case,  the  process  described  in  this  article  will  be  unnecessary 


NOTE  TO  ARTICLE  462. 

Platform  stairs  viith  a  large  cylinder.  Instead  of 
placing  ihe  platform-risers  at  the  spring  of  the  cyl- 
inder, a  more  easy  and  graceful  appearance  may  be  • 
given  to  the  rail,  and  the  necessity  of  canting  either 
of  the  twists  entirely  obviated,  by  fixing  the  place  of 
the  above  risers  at  a  certain  distance  within  the  cyl- 
inder, as  shown  in  the  annexed  cut — the  lines  indi- 
cating the  face  of  the  risers  cutting  the  cylinder  at  k 
and  Z,  instead  of  at  p  and  q,  the  spring  of  the  cylin- 
der. To  ascertain  the  position  of  the  risers,  let  a  6  c 
be  the  pitch-board  of  the  lower  flight,  and  c  d  e  that 
of  the  upper  flight,  these  being  placed  so  that  6  c 
and  c  d  shall  form  a  right  line.  Extend  a  c  to  cut 
d  e  in  f ;  draw  f  g  parallel  to  d  b,  and  of  indefinite 
length :  draw  g  o  at  right  angles  to  /  g,  and  equal 
in.  length  to  the  radius  of  the  circle  formed  by  the 
centre  of  the  rail  in  passing  around  the  cylinder ; 
on  o  as  centre  describe  the  semicircle  j  g  i;  make 
o  h  equal  to  the  radius  of  the  cylinder,  and  describe 
on  o  the  face  of  the  cylinder  p  h  q  ;  then  extend  d  b 
across  the  cylinder,  cutting  it  in  Z  and  k — giving  the 
position  of  the  face  of  the  risers,  as  required.  To 
find  the  face-mould  for  the  twists  is  simple  and  ob- 
vious :  it  being  merely  a  quarter  of  an  ellipse,  hav- 
ing o  j  for  semi-minor  axis,  and  the  distance  on  the 

rake  corresponding  to  o  g,  on  the  plan,  for  the  semi-major  axis,  found  thus, — extend  i  j 
meet  a  f,  then  from  this  point  of  meeting  to  f  is  the  semi-major  axis. 


SECTION  VII.— SHADOWS. 


499. — The  art  of  drawing  consists  in  representing  solids  upon 
a  plane  surface  :  so  that  a  curious  and  nice  adjustment  of  lines  ia 
n  ade  to  present  the  same  appearance  to  the  eye,  as  does  the 
human  figure,  a  tree,  or  a  house.  It  is  by  the  effects  of  light,  in 
its  reflection,  shade,  and  shadow,  that  the  presence  of  an  object  is 
made  known  to  us ;  so,  upon  paper,  it  is  necessary,  in  order  that 
the  delineation  may  appear  real,  to  represent  fully  all  the  shades 
and  shadows  that  would  be  seen  upon  the  object  itself.  In  this 
section  I  propose  to  illustrate,  by  a  few  plain  examples,  the  simple 
elementary  principles  upon  which  shading,  in  architectural  sub- 
jects, is  based.  The  necessary  knowledge  of  drawing,  prelim- 
inary to  this  subject,  is  treated  of  in  the  Introduction,  from  Art. 
1  to  14. 

500. —  Tlie  inclination  of  the  line  of  shadow.  This  is  always, 
in  architectural  drawing,  45  degrees,  both  on  the  elevation  and  the 
plan  ;  and  the  sun  is  supposed  to  be  behind  the  spectator,  and 
over  his  left  shoulder.  This  can  be  illustrated  by  reference  to 
Fig.  335,  in  which  A  represents  a  horizontal  plane,  and  B  and  C 
two  vertical  planes  placed  at  right  angles  to  each  other.  A  rep- 
resents the  plan,  C  the  elevation,  and  B  a  vertical  projection 
from  the  elevation.  In  finding  the  shadow  of  th  3  plane,  B,  the 


SHADOWS. 


Fig.  885. 

line,  a  b,  is  drawn  at  an  angle  of  45  degrees  with  the  horizon,  and 
the  line,  c  b,  at  the  same  angle  with  the  vertical  plane,  B.  The 
plane,  B,  being  a  rectangle,  this  makes  the  true  direction  of  the 
sun's  rays  to  be  in  a  course  parallel  to  d  b ;  which  direction  has 
been  proved  to  be  at  an  angle  of  35  degrees  and  16  minutes  with 
the  horizon.  It  is  convenient,  in  shading,  to  have  a  set-square 
with  the  two  sides  that  contain  the  right  angle  of  equal  length  ; 
this  will  make  the  two  acute  angles  each  45  degrees ;  and  will 
give. the  requisite  bevil  when  worked  upon  the  edge  of  the  T- 
square.  One  reason  why  this  angle  is  chosen  in  preference  to 
another,  is,  that  when  shadows  are  properly  made  upon  the  draw- 
ing by  it,  the  depth  of  every  recess  is  more  readily  known,  since 
the  breadth  of  shadow  and  the  depth  of  the  recess  will  be  equal. 

To  distinguish  between  the  terms  shade  and  shadow,  it  will  be 
understood  that  all  such  parts  of  a  body  as  art  not  exposed  to  the 
direct  action  of  the  sun's  rays,  are  in  shade ;  while  those  parts 
which  are  deprived  of  light  by  the  interposition  of  other  bodies, 
are  in  shadow. 


384: 


AMERICAN    HOUSE-CARPENTER. 


501. —  To  find  the  line  of  shadow  on  mouldings  and  other  ho- 
rizontally straight  projections..  Fig.  336,  337,  338,  and  339, 
lepresent  various  mouldings  in  elevation,  returned  at  the  left,  in 
the  usual  manner  of  mitreing  around  a  projection.  A  mere  in- 
spection of  the  figures  is  sufficient  to  see  how  the  line  of  shadow 
is  obtained  ;  bearing  in  mind  that  the  ray,  a  b,  is  drawn  from  the 
projections  at  an  angle  of  45  degrees.  Where  there  is  no  return 
at  the  end,  it  is  necessary  to  draw  a  section,  at  any  place  in  the 
length  of  the  mouldings,  and  find  the  line  of  shadow  from  that. 

502. — To  find  the  line  of  shadow  cast  by  a  shelf.  In  Fig.  340, 
A  is  the  plan,  and  B  is  the  elevation  of  a  shelf  attached  to  a  wall. 
From  a  and  c,  draw  a  b  and  c  d,  according  to  the  angle  previously 
directed  ;  from  b,  erect  a  perpendicular  intersecting  c  d  at  d ;  from 
d,  draw  d  e,  parallel  to  the  shelf ;  then  the  lines,  c  d  and  d  e,  will 
define  the  shadow  cast  by  the  shelf.  There  is  another  method  of 
finding  the  shadow,  without  the  plan,  A.  Extend  the  lower  line 
of  tne  shelf  toy,  and  make  cf  equal  to  the  projection  of  the  shelf 


SHADOWS. 


385 


from  the  wall ;  from/,  draw/^,  at  the  customary  angle,  and  from 
c,  drop  the  vertical  line,  c  g,  intersecting  /  g  at  g  ;  from  g,  draw 
g  e,  parallel  to  the  shelf,  and  from  c,  draw  c  d,  at  the  usual  angle  ; 
then  the  lines,  c  d  and  d  e,  will  determine  the  extent  of  the  shadow 
as  before. 


Fig.  84L 


503. — To  find  the  shadow  cast  by  a  shelf ,  which  is  wider  at 
one  end  than  at  the  other.  In  Fig.  341,  A  is  the  pJan,  and  B 
the  elevation.  Find  the  point,  d,  as  in  the  previous  example,  and 
from  any  other  point  in  the  front  of  the  shelf,  as  a,  erect  the  perpen- 
dicular, a  e ;  from  a  and  e,  draw  a  b  and  e  c,  at  the  proper  angle, 
and  from  b,  erect  the  perpendicular,  b  c,  intersecting  e  c  in  c  ; 
49 


386 


AMERICAN    HOUSE-CARPENTER. 


from  d,  through  c,  draw  d  o  ;  then  the  lines,  *  d  and  d  o,  will  giro 
the  limit  of  the  shadow  cast  by  the  shelf. 


Fig.  842. 

504:. — To  find  the  shadow  of  a  shelf  having  one  end  acute  or 
obtuse  angled.  Fig.  342  shows  the  plan  and  elevation  of  an 
acute-angled  shelf.  Find  the  line,  e  g,  as  before  ;  from  a,  erect 
the  perpendicular,  a  b  •  join  6  and  e ;  then  b  e  and  e  g  will  define 
the  boundary  of  shadow. 


Fig.  848. 

505. — To  find  the  shadow  cast  by  an  inclined  shelf.  In  Fig. 
343,  the  plan  and  elevation  of  such  a  shelf  is  shown,  having  also 
one  end  wider  than  the  other.  Proceed  as  directed  for  finding 
the  shadows  of  Fig.  341,  and  find  the  points,  d  and  c  ;  then  a  d 
an  I  d  c  will  be  the  shadow  required.  If  the  shelf  had  been 


SHADOWS. 


38  't 


parallel  in  width  on  the  plan,  then  the  line,  d  c,  would  have  beei 
parallel  with  the  shelf,  a  b. 


Fig.  344 


Fig.  845. 


506. — To  find  the  shadow  cast  by  a  shelf  inclined  in  its  ver 
tical  section  either  upward  or  downward.  From  a,  (Fig.  3-i- 
and  345,)  draw  a  b,  at  the  usual  angle,  and  from  b,  draw  b  c 
parallel  with  the  shelf ;  obtain  the  point,  e,  by  drawing  a  lin 
from  d,  at  the  usual  angle.  In  Fig.  344,  join  e  and  i ;  then  i 
and  e  c  will  define  the  shadow.  In  Fig.  345,  from  o,  draw  o  \ 
parallel  with  the  shelf;  join  i  and  e ;  then  i  e  and  e  c  will  be  th< 
shadow  required. 

The  projections  in  these  several  examples  are  bounded  b; 
straight  lines ;  but  the  shadows  of  curved  lines  may  be  found  i: 
the  same  manner,  by  projecting  shadows  from  several  points  \\ 
the  curved  line,  and  tracing  the  curve  of  shadow  through  thesi 
points.  Thus — 


Fig.  S47 


JTg.  846. 


OOO  AMERICAN    HOLSE  CARPENTER. 

507- — To  find  the  shadow  of  a  shelf  having  its  front  edge,  of 
end,  curved  on  the  plan.  In  Fig.  346  and  347,  A  and  A  show  an 
example  of  each  kind.  From  several  points,  as  a,  a,  in  the  plan, 
and  from  the  corresponding  points,  o,  o,  in  the  elevation,  draw 
rays  and  perpendiculars  intersecting  at  e,  e,  &c. ;  through  these 
points  of  intersection  trace  the  curve,  and  it  will  define  the  shadow. 


Fig.  848. 


508. — To  find  the  shadow  of  a  shelf  curved  in  the  elevation. 
In  Fig.  348,  find  the  points  of  intersection,  e,  e  and  e,  as  in  the 
last  examples,  and  a  curve  traced  through  them  will  define  the 
shadow. 

The  preceding  examples  show  how  to  find  shadows  when  cast 
upon  a  vertical  plane ;  shadows  thrown  upon  curved  surfaces  are 
ascertained  in  a  similar  manner.  Thus — 


Flf.  849. 


SHADOWS. 


509. —  To  find  the  shadow  cast  upon  a  cylindrical  wall  by  a 
projection  of  any  kind.  By  an  inspection  of  Fig.  349,  it  will  be 
seen  that  the  only  difference  between  this  and  the  last  examples, 
is,  that  the  rays  in  the  plan  die  against  the  circle,  a  b,  instead  of 
a  straight  line. 


Fig.  sea 

510. —  To  find  the  shadow  cast  by  a  shelf  upon  an  inclined 
wall.  Cast  the  ray,  a  b,  (Fig.  350,)  from  the  end  of  the  shelf  to 
the  face  of  the  wall,  and  from  b,  draw  b  c,  parallel  to  the  shelf; 
cast  the  ray,  d  e,  from  the  end  of  the  shelf;  then  the  lines,  d  e 
and  e  c,  will  define  the  shadow. 

These  examples  might  be  multiplied,  but  enough  has  been 
given  to  illustrate  the  general  principle,  by  which  shadows  in  all 
instances  are  ibund.  Let  us  attend  now  to  the  application  of  this 
principle  to  such  familiar  objects  as  are  likely  to  occur  in  practice. 


Fig.  85L 


390 


AMERICAN    HOUSE-CARPENTER. 


511. —  To  Jind  the  shadow  of  a  projecting  horizontal  beam 
From  the  points,  a,  a,  &c.,  (Fig.  351,)  cast  rays  upon  the  wall ; 
the  intersections,  e,  e,  e,  of  those  rays  with  the  perpendiculars 
drawn  from  the  plan/will  define  the  shadow.  If  the  beam  be  in- 
clined, either  on  the  plan  or  elevation,  at  any  angle  other  than  a 
right  angle,  the  difference  m  the  manner  of  proceeding  can  be  seen 
by  reference  to  the  preceding  examples  of  inclined  shelves  &c. 


Fig.  862. 

512. — To  Jind  the  shadow  in  a  recess.  From  the  point,  a, 
(Fig.  352,)  in  the  plan,  and  b  in  the  elevation,  draw  th  •  rays,  a  c 
and  b  e ;  from  c,  erect  the  perpendicular,  c  e,  and  fn,ni  e,  draw 
the  horizontal  line,  e  d ;  then  the  lines,  c  e  and  e  d,  will  show  the 
extent  of  the  shadow.  This  applies  only  where  the  back  of  the 
recess  is  parallel  with  the  face  of  the  wall. 


Fig.  858. 


513. — To  find  the  shadow  in  a  recess,  when  the  face  of  the 
wall  is  inclined,  and  the  back  of  the  recess  is  vertical.  In  Fig. 
353,  A  shows  the  section  and  B  the  deration  of  a  recess  of  this 


SHADOWS.  391 

kind.  From  b,  and  from  any  other  point  in  the  line,  b  a  as  a 
draw  the  rays,  b  c  and  a  e ;  from  c,  a,  and  e,  draw  the  hcrizonta' 
lines,  c  gt  a  ft  and  eh;  from  d  and  /,  cast  the  rays,  d  i  and/  h  • 
from  i,  through  h,  draw  i  s  ;  then  5  i  and  i  ^  will  define  tht 
shadow. 

id 


Fig.  854. 


514. — To  find  the  shadow  in  a  fireplace.  From  a  and  fc, 
(Fig.  354,)  cast  the  rays,  a  c  and  b  e,  and  from  c,  erect  the  per- 
pendicular, c  e ;  from  e,  draw  the  horizontal  line,  e  o,  and  join  r 
and  d ;  then  c  e,  e  o,  and  o  d,  will  give  the  extent  of  the  shadow. 


Fig.  856. 


515. — To  find  the  shadow  of  a  moulded  window-lintel.  Cast 
rays  from  the  projections,  a,  c,  & ;.,  in  the  plan,  (Fig.  355,)  and 
d,  e,  &c.,  in  the  elevation,  anc  draw  the  usual  perpendiculars  in- 
tersecting the  rays  at  t,  i,  and  i ;  these  intersections  connected 


392 


AMERICAN    HOUSE-CARPENTER. 


and  horizontal  lines  drawn  frorr  them,  will  define  the  shadow 
The  shadow  on  the  face  of  the  lintel  is  found  by  casting  a  rai 
back  from  i  to  s,  and  drawing  the  horizontal  line,  s  n. 


Fig.  856. 

516. —  To  find  the  shadow  cast  by  the  nosing  of  a  step.  From 
«,  (Fig.  356,)  and  its  corresponding  point,  c,  cast  the  rays,  a  b 
and  c  d,  and  from  b,  erect  the  perpendicular,  b  d ;  tangical  to  the 
curve  at  e,  cast  the  ray,  e  f,  .and  from  e,  drop  the  perpendicular, 
e  o,  meeting  the  mitre-line,  a  g,  in  o ;  cast  a  ray  from  o  to  i,  and 
from  i,  erect  the  perpendicular,  i  f ;  from  h,  draw  the  ray,  h  k; 
from  f  to  d  and  from  d  to  k,  trace  the  curve  as  shown  in  the 
figure  ;  from  k  and  h,  draw  the  horizontal  lines,  k  n  and  h  s ,"  then 
the  limit  of  the  shadow  will  be  completed. 

517. —  To  find  the  shadow  thrown  by  a  pedestal  upon  steps. 
From  a,  (Fig.  357,)  in  the  plan,  and  from  c  in  the  elevation,  draw 
the  rays,  a  b  and  c  e ;  then  a  o  will  show  the  extent  of  the  shadow 
on  the  first  riser,  as  at  A  ;  f  g  will  determine  the  shadow  on  the 
second  riser,  as  at  B ;  c  d  gives  the  amount  of  shadow  on  the 
first  tread,  as  at  C,  and  h  i  that  on  the  second  tread,  as  at  D ; 
which  completes  the  shadow  of  the  left-hand  pedestal,  both  on  the 
plar  and  elevation.  A  mere  inspection  of  the  figure  will  be  suf- 


SHADOWS. 


393 


Fig.  867. 

ficient  to  show  how  the  shadow  of  the  right-hand  pedestal  is 
obtained. 


Fig.  869. 

518. — To  find  the  shadow  thrown  on  a  column  by  a  square 
abacus.  From  a  and  b,  (Fig.  358,)  draw  the  rays,  a  c  and  b  e, 
and  from  c,  erect  the  perpendicular,  c  e  ;  tangical  to  the  curve  at 
d,  draw  the  ray,  d  /,  and  from  h,  corresponding  to  /  in  the  plan, 
draw  the  ray,  ho;  take  any  point  between  a  and/,  as  i,  and  from 
this,  as  also  from  a  corresponding  point,  n,  draw  the  rays,  i  r  and 
n  s ;  from  r,  and  from  d,  erect  the  perpendiculars,  r  s  and  do; 
through  the  points,  e,  s,  and  o,  trace  the  curve  as  shown  in  the 
figure  ;  then  the  extent  of  the  shadow  will  be  defined. 

519. — To  find  the  shadow  thrown  on  a  column,  by  a  circular 
abacus.     This  is  so  near  like  the  iast  example,  that  no  explanation 
will  be  necessary  farther  than  a  reference  to  the  preceding  article 
50 


394 


AMERICAN    HOUSE-CARPEJSTER. 


m 

x 

\  /" 

\A 

A.   f 

\^                             ^S*^x 

Fig.  860. 

520. — To  find  the  shadows  on  the  capital  of  a  column.  This 
may  be  done  according  to  the  principles  explained  in  the  examples 
already  given ;  a  quicker  way  >f  doing  it,  however,  is  as  follows 
If  we  take  into  consideration  one  ray  of  light  in  connection  with 
all  those  perpendicularly  under  and  over  it,  it  is  evident  that  these 
several  rays  would  form  a  vertical  plane,  standing  at  an  angle  ol 
45  degrees  with  the  face  of  the  elevation.  Now,  we  may  sup 
pose  the  column  to  be  sliced,  so  to  speak,  with  planes  of  this 


395 


Fig.  861. 

nature — cutting  it  in  the  lines,  a  b,  c  d,  &c.,  (Fig.  360,)  and,  in 
the  elevation,  find,  by  squaring  up  from  the  plan,  the  lines  of  sec- 
tion which  these  planes  would  make  thereupon.  For  instance  : 
in  finding  upon  the  elevation  the  line  of  section,  a  b,  the  plane 
cuts  the  ovolo  at  e,  and  therefore  /  will  be  the  corresponding  point 
upon  the  elevation  ;  h  corresponds  with  g,  i  with  j,  o  with  s,  and 
I  with  b.  Now,  to  find  the  shadows  upon  this  line  of  section,  cast 
from  m,  the  ray,  m  n,  from  h,  the  ray,  h  o,  &c. ;  then  that  part  of 
the  section  indicated  by  the  letters,  m  f  i  n,  and  tjiat  part  also  be- 
tween h  and  o,  will  be  under  shadow.  By  an  inspection  of  the 
figure,  it  will  be  seen  that  the  same  process  is  applied  to  each  line 
of  section,  and  in  that  way  the  points,  p,  r,  t,  u,  v,  tv,  x,  as  alsc 
1,  2,  3,  &c.,  are  successively  found,  and  the  lines  of  shadow 
traced  through  them. 

Fig.  361  is  an  example  of  the  same  capital  with  all  the  shadows 
finished  in  accordance  with  the  lines  obtained  on  Fig.  360. 

521. — To  find  the  shadow  thrown  on  a  vertical  wall  by  a 
column  and  entablature  standing  'n  advance  of  said  wall.  Cast 


396 


AMERICAN    HOUSE-CARPENTER 


Fig.  862. 


rays  from  a  and  b,  (Fig.  362,)  and  find  the  point,  e,  as  in  the 
previous  examples ;  from  d,  draw  the  ray,  d  e,  and  from  e,  the 
horizontal  line,  e  f ;  tangical  to  the  curve  at  g  and  h,  draw  the 
rays,  g  j  and  h  i,  and  from  i  and  j,  erect  the  perpendiculars,  i  I 
and^'  k ;  from  m  and  n,  draw  the  rays,  m/and  n  k,  and  trace  the 
curve  between  k  and  f;  cast  a  ray  from  o  to  p,  a  vertical  line 
from  p  to  *,  and  through  s,  draw  the  horizontal  line,  *  t ;  the 
shadow  as  required  will  then  be  completed. 


397 


Fig.  S63. 


Fig.  363  is  an  example  of  the  same  kind  as  the  last,  with  all 
ihe  shadows  filled  in,  according  to  the  lines  obtained  in  the  pre- 
ceding figure. 


Fig.  864. 


522  —Fig.  364  and  365  are  examples  of  the  Tuscan  cornice. 
The  manner  of  obtainir  g  the  shadows  is  evident. 


398 


AMERICAN    HOUSE-CARPENTER. 


Kg.  865. 

523. — Reflected  light.  In  shading,  the  finish  and  life  of  ar* 
object  depend  much  on  reflected  light  This  is  seen  to  advantage 
in  Fig.  361  and  on  the  column  in  Fig.  363.  Reflected  rays  are 
thrown  in  a  direction  exactly  the  reverse  of  direct  rays  ;  therefore, 
on.  that  part  of  an  object  which  is  subject  to  reflected  light,  the 
shadows  are  reversed.  The  fillet  of  the  ovolo  in  Fig.  361  is  an 
example  of  this.  On  the  right-hand  side  of  the  column,  the  face 
of  the  fillet  is  much  darker  than  the  cove  directly  under  it.  The 
reason  of  this  is,  the  face  of  the  fillet  is  deprived  both  of  direct 
and  reflected  light,  whereas  the  cove  is  subject  to  the  latter. 
Other  instances  of  the  effect  of  reflected  light  will  te  seen  in  the 
other  examples. 


APPENDIX, 


ALGEBKAICAL  SIGNS. 


4-,  plus,  signifies  addition,  and  that  the  two  quantities  between  which 
it  stands  are  to  be  added  together ;  as  a  +  b,  read  a  added  to  b. 

— ,  minus,  signifies  subtraction,  or  that  of  the  two  quantities  between 
which  it  occurs,  the  latter  is  to  be  subtracted  from  the  former ;  as 
a  —  b,  read  a  minus  b. 

X ,  multiplied  by,  or  the  sign  of  multiplication.  It  denotes  that  the  two 
quantities  between  which  it  occurs  are  to  be  multiplied  together ; 
as  a  x  b,  read  a  multiplied  by  b,  or  a  times  b.  This  sign  is  usually 
omitted  between  symbols  or  letters,  and  is  then  understood,  as  ab. 
This  has  the  same  meaning  as  a  x  b.  It  is  never  omitted  between 
arithmetical  numbers ;  as  9  X  5,  read  nine  times  five. 

— ,  divided  by,  or  the  sign  of  division,  and  denotes  that  of  the  two  quanti- 
ties between  which  it  occurs,  the  former  is  to  be  divided  by  the  latter ; 
as  a  —  b,  read  a  divided  by  6.  Division  is  also  represented  thus  : 

-,      in  the  form  of  a  fraction.     This  signifies  that  a  is  to  be  divided  by 

6       6.     When  more  than  one  symbol  occurs  above  or  below  the  line, 

or  both,  as  ^.T,  it  denotes  that  the  product  of  the  symbols  above 

the  line  is  to  be  divided  by  the  product  of  those  below  the  line. 

=,  is  equal  to,  or  sign  of  equality,  and  denotes  that  the  quantity  0 
quantities  on  its  left  are  equal  to  those  on  its  right;  as  a  -  b  _  c, 
read  a  minus  6  is  equal  to  c,  or  equals  c  ;  or,  9  —  5  =  4,  read  nine 
minus  five  equals  four.     This  sign,  together  with  the  symbols  on 
each  side  of  it,  when  spoken  of  as  a  whole,  is  called  an  equation. 

a*  denotes  a  squared,  or  a  multiplied  by  a,  or  the  second  pofl 

a'  denotes  a  cubed,  or  a  multiplied  by  a  and  again  multiplied  by  a,  or 
the  third  power  of  a.     The  small  figure,  2,  3,  or  4,  &c.,  is  tenr 
the  index  or  exponent  of  the  power.     It  indicates  how  many  times 
the  symbol  is  to  be  taken.     Thus,  c?  =  aa,a*=aa a  a  =  a  a  a  a 
/  is  the  radical  sign,  and  denotes  that  the  square  root  of  the  quai 
following  it  is  to  be  extracted,  and 
51 


4  APPENDIX. 

•v/  denotes  that  the  cube  root  of  the  quantity  following  it  is  to  be  ex- 
tracted. Thus,  V9  =  3,  and  -^27  =  3.  The  extraction  of  root* 
is  also  denoted  by  a  fractional  index  or  exponent^  thus 

a*  denotes  the  square  root  of  a, 

a**  denotes  the  cube  root  of  a, 

a*  denotee  the  cube  root  of  the  square  of  a,  &c. 


TRIGONOMETRICAL  TERMS. 


Ffg.8«<L 


In  Fig.  366,  where  AB  is  the  radius  of  the  circle  B  Off,  draw  a 
line  A  F,  from  A,  through  any  point,  (7,  of  the  arc  B  G.  From  C  draw 
CD  perpendicular  to  A  B  ;  from  B  draw  B  E  perpendicular  to  A  B  ; 
and  from  G  draw  G  F  perpendicular  to  A  G. 

Then,  for  the  angle  F  A  B,  when  the  radius  A  C  equals  unity,  CD 
is  the  sine  ;  A  D  the  cosine  ;  D  B  the  versed  sine  ;  B  E  the  tangent , 
GF  the  cotangent;  AE  the  secant;  amdAF  the  cosecanf. 


6 


APPENDIX. 


But  if  the  angle  be  larger  than  one  right  angle,  yet  less  than  two 
right  angles,  as  B  A  H,  extend  HA  to  K  and  EB  to  K,  and  from  H 
draw  H  J  perpendicular  to  AJ. 

Then,  for  the  angle  BAH,  when  the  radius  A  H  equals  unity,  HJ  is 
the  sine  ;  AJ  the  cosine  ;  BJ  the  versed  sine  ;  B  K  the  tangent ;  and 
A  K  the  secant. 

When  the  number  of  degrees  contained  in  a  given  angle  is  known, 
then  the  value  of  the  sine,  cosine,  <fcc.,  corresponding  to  that  angle,  may 
be  found  in  a  table  of  Natural  Sines,  Cosines,  &c. 

In  the  absence  of  such  a  table,  and  when  the  degrees  contained  in  the 
given  angle  are  unknown,  the  values  of  the  sine,  cosine,  &C.,  may 
be  found  by  computation,  as  follows: — Let  B  A  (7,  (Fig.  367,)  be  the 


Fig.  867. 


given  angle.  At  any  distance  from  A,  draw  c  perpendicular  to  A  B. 
By  any  scale  of  equal  parts  obtain  the  length  of  each  of  the  three  linea 
a,  6,  c.  Then  for  the  angle  at  A  we  have,  by  proportion, 


1-0 
1-0 
1-0 


sin.  =  -. 
a 


tan.  =  7. 
o 


1-0    :    cot. 


a  ::  i-o 


1-0 


a 

cosec.  =  - . 
c 


Or,  in  any  right  angled  triangle,  for  the  angle  contained  between  the 
base  and  hypothenuse — 

Whec  perp.  divided  by  hyp.,  the  quotient  equals  the  sine. 


£ 
base 

«  perp. 

«  base 

"  hyp. 

«  hyp. 


perp.,  " 
base,.  " 
perp.,  " 


cosine. 

tangent. 

cotangent. 

secant. 

Cosecant. 


GLOSSARY. 


not  found  here  can  be  found  in  the  lists  of  definitions  in  other  parts  of  :hia  book, 
or  in  common  dictionaries. 


Abacus. — The  uppermost  member  of  a  capital. 

Abbatoir. — A  slaughter-house. 

Abbey. — The  residence  of  an  abbot  or  abbess. 

Abutment. — That  part  of  a  pier  from  which  the  arch  springs. 

Acanthus. — A  plant  called  in  English,  bear's-breech.  Its  leaves  arc 
employed  for  decorating  the  Corinthian  and  the  Composite  capitals. 

Acropolis. — The  highest  part  of  a  city  ;  generally  the  citadel. 

Acroteria. — The  small  pedestals  placed  on  the  extremities  and  apex 
ov  a  pediment,  originally  intended  as  a  base  for  sculpture. 

Aisle. — Passage  to  and  from  the  pews  of  a  church.  In  Gothic  ar- 
chitecture, the  lean-to  wings  on  the  sides  of  the  nave. 

Alcove. — Part  of  a  chamber  separated  by  an  estrade,  or  partition  of 
columns.  Recess  with  seats,  &c.,  in  gerdens. 

Altar. — A  pedestal  whereon  sacrifice  was  offered.  In  modern 
churches,  the  area  within  the  railing  in  front  of  the  pulpit. 

Alto-relievo. — High  relief;  sculpture  projecting  from  a  surface  so  as 
to  appear  nearly  isolated. 

Amphitheatre  — A  double  theatre,  employed  by  the  ancients  for  the 
exhibition  of  giadmtorial  fights  and  other  shows. 

Ancones. — Trusses  employed  as  an  apparent  support  to  a  cornice 
upon  the  flanks  or*  the  architrave. 

Annulet. — A  small  square  moulding  used  to  separate  others;  the 
filLets  in  the  Doric  capital  under  the  ovolo,  and  those  which  separate 
the  flutings  of  columns,  are  known  by  this  term. 

Antce. — A  pilaster  attached  to  a  wall. 

Apiary. — A  place  for  keeping  beehives. 

Arabesque. — A  building  after  the  Arabian  style. 

Areostyle. — An  intercolumniation  of  from  four  to  five  diameters. 

Arcade — A  series  of  arches. 

Arch. — An  arrangement  of  stones  or  other  material  in  a  curvilinear 
form,  so  as  to  perform  the  office  of  a  lintel  and  carry  superincumbent 
weights. 

Architrave. — That  part  of  the  entablature  whbh  rests  upon  the 
capital  of  a  column,  and  is  beneath  the  frieze.  The  casing  and 
mouldings  about  a  door  or  window. 


APPENDIX. 

Archivolt. — The  ceiling  of  a  vault :  the  under  surface  of  an  aich. 

Area. — Superficial  measurement.  An  open  space,  below  the  leve: 
of  the  ground,  in  front  of  basement  windows. 

Arsenal. — A  public  establishment  for  the  deposition  of  arms  and 
warlike  stores. 

Astragal. — A  email  moulding  consisting  of  a  half-round  with  a  fillet 
on  each  side. 

Attic. — A  low  story  erected  over  an  order  of  architecture.     A  low 
additional  story  immediately  under  the  roof  of  a  building. 
-Aviary. — A  place  for  keeping  and  breeding  birds. 

Balcony. — An  open  gallery  projecting  from  the  front  of  a  building. 

Baluster. — A  small  pillar  or  pilaster  supporting  a  rail. 

Balustrade. — A  series  of  balusters  connected  by  a  rail. 

Barge-course. — That  part  of  the  covering  which  projects  over  the 
gable  of  a  building. 

Base. — The  lowest  part  of  a  wall,  column,  &c. 

Basement-story. — That  which  is  immediately  under  the  principal 
story,  and  included  within  the  foundation  of  the  building. 

Basso-relievo. — Low  relief;  sculptured  figures  projecting  from  a 
surface  one-half  their  thickness  or  less.  See  Alto-relievo. 

Battering. — See  Talus. 

Battlement. — Indentations  on  the  top  of  a  wall  or  parapet. 

Bay-window. — A  window  projecting  in  two  or  more  planes,  and  not 
forming  the  segment  of  a  circle. 

Bazaar. — A  species  of  mart  or  exchange  for  the  sale  of  various  ar- 
ticles of  merchandise. 

Bead. — A  circular  moulding. 

Bed-mouldings. — Those  mouldings  which  are  between  the  corona 
and  the  frieze. 

Belfry. — That  part  of  a  steeple  in  which  the  bells  are  hung  :  an- 
ciently called  campanile. 

Belvedere. — An  ornamental  turret  or  observatory  commanding  a 
pleasant  prospect. 

Bow-window. — A  window  projecting  in  curved  lines. 

Bressummer. — Abeam  or  iron  tie  supporting  a  wall  over  a  gateway 
or  other  opening. 
^Brick-nagging. — The  brickwork  between  studs  of  partitions. 

Buttress. — A  projection  from  a  wall  to  give  additional  strength. 

Cable. — A  cylindrical  moulding  placed  in  flutes  at  the  lower  part  of 
the  column. 

Camber. — To  give  a  convexity  to  the  upper  surface  of  a  beam. 

Campanile. — A  tower  for  the  reception  of  bells,  usually,  in  Italy, 
separated  from  the  church. 

Canopy. — An  ornamental  covering  over  a  seat  ot  state. 

Canfalivers. — The  ends  of  rafters  under  a  projecting  roof.  Pieces 
of  wood  or  stone  supporting  the  eaves. 

Capital. — The  uppermost  part  of  a  column  included  between  the 
shaft  and  the  architrave. 


APPENDIX.  9 

Caravansera. — In  the  East,  a  large  public  building  for  tiie  reception 
of  travellers  by  caravans  in  the  desert. 

Carpentry. — (From  the  Latin,  carpenium,  carved  wood.)  That  de- 
partment of  science  and  art  which  treats  of  the  disposition,  the  con. 
struction  and  the  relative  strength  of  timber.  Th^  first  is  called  de. 
scriptive,  the  second  constructive,  and  the  last  mechanical  carpentry. 

Caryatides. — Figures  of  women  used  instead  of  columns  to  support 
an  entablature. 

Casino. — A  small  country-house. 

Castellated. — Built  with  battlements  and  turrets  in  imitation  of  an- 
cient  castles. 

Castle. — A  building  fortified  for  military  defence.  A  house  with 
towers,  usually  encompassed  with  walls  and  moats,  and  having  a  don- 
jon,  or  keep,  in  the  centre. 

Catacombs. — Subterraneous  places  for  burying  the  dead. 

Cathedral. — The  principal  church  of  a  province  or  diocese,  wherein 
the  throne  of  the  archbishop  or  bishop  is  placed. 

Cavetto. — A  concave  moulding  comprising  the  quadrant  of  a  circle. 

Cemetery. — An  edifice  or  area  where  the  dead  are  interred. 

Cenotaph. — A  monument  erected  to  the  memory  of  a  person  buried 
in  another  place. 

Centring. — The  temporary  woodwork,  or  framing,  whereon  any 
vaulted  work  is  constructed. 

Cesspool. — A  well  under  a  drain  or  pavement  to  receive  the  waste- 
water  and  sediment. 

Chamfer. — The  bevilled  edge  of  any  thing  originally  right-angled. 

Chancel. — That  part  of  a  Gothic  church  in  which  the  altar  is  placed. 

Chantry. — A  little  chapel  in  ancient  churches,  with  an  endowment 
for  one  or  more  priests  to  say  mass  for  the  relief  of  souls  out  of  purga- 
tory. 

Chapel. — A  building  for  religious  worship,  erected  separately  from 
a  church,  and  served  by  a  chaplain. 

Chapht. — A  moulding  carved  into  beads,  olives,  &c. 

Cincture. — The  ring,  listel,  or  fillet,  at  the  top  and  bottom  of  a  co- 
lumn, which  divides  the  shaft  of  the  column  from  its  capital  and  base. 

Circus. — A  straight,  long,  narrow  building  used  by  the  Romans  for 
the  exhibition  of  public  spectacles  and  chariot  races.  At  the  present 
day,  a  building  enclosing  an  arena  for  the  exhibition  of  feats  of  horse, 
manship. 

Clere-story. — The  upper  part  of  the  nave  of  a  church  above  the 
roofs  of  the  aisles. 

Cloister. — The  square  space  attached  to  a  regular  monastery  o? 
large  church,  having  a  peristyle  or  ambulatory  around  it,  covered  with 
a  range  of  buildings. 

Coffer-dam. — A  case  of  piling,  water-tight,  fixed  in  the  bed  of  a 
river,  for  the  purpose  of  excluding  the  water  while  any  work,  such  as 
a  wharf,  wall,  or  the  pier  of  a  bridge,  is  carried  up. 

Collar-learn. — A  horizontal  beam  fra  ncd  betw-ecn  two  principal 
rafters  above  the  tie-beam. 

Cullcnade. — A  range  of  columns. 

Columbarium. — A  pigeon-house. 


10 


APPENDIX. 


Column. — A  vertical,  cylindrical  support  under  the  entablature  of 
an  order. 

Common-rafters. — The  same  as  jack-rafters,  which  see 

Conduit. — A  long,  narrow,  walled  passage  underground,  for  secret 
communication  between  different  apartments.  A  canal  or  pipe  for  the 
conveyance  of  water. 

Conservatory. — A  building  for  preserving  curious  and  rare  exotio 
plants. 

Consoles. — The  same  as  ancones,  which  see. 

Contour. — The  external  lines  which  bound  and  terminate  a  figure. 

Convent. — A  building  for  the  reception  of  a  society  of  religious  per- 
sons. 

Coping. — Stones  laid  on  the  top  of  a  wall  to  defend  it  from  the 
weather. 

Corbels. — Stones  or  timbers  fixed  in  a  wall  to  sustain  the  timbers  of 
a  floor  or  roof. 

Cornice. — Any  moulded  projection  which  crowns  or  finishes  the 
part  to  which  it  is  affixed. 

Corona. — That  part  of  a  cornice  which  is  between  the  crown- 
moulding  and  the  bed-mouldings. 

Cornucopia. — The  horn  of  plenty. 

Corridor. — An  open  gallery  or  communication  to  the  different  apart- 
ments of  a  house. 

Cove. — A  concave  moulding. 

Cripple-rafters. — The  short  rafters  which  are  spiked  to  the  hip-rafter 
of  a  roof. 

Crockets. — In  Gothic  architecture,  the  ornaments  placed  along  the 
angles  of  pediments,  pinnacles,  &c. 

Crosettes. — The  same  as  ancones,  which  see. 

Crypt. — The  under  or  hidden  part  of  a  building. 

Culvert. — An  arched  channel  of  masonry  or  brickwork,  built  be- 
neath the  bed  of  a  canal  for  the  purpose  of  conducting  water  under  it. 
\ny  arched  channel  for  water  underground. 

Cupola. — A  small  building  on  the  top  of  a  dome. 

Cur tail -step. — A  step  with  a  spiral  end,  usually  the  first  of  the  flight. 

Cusps. — The  pendents  of  a  pointed  arch. 

Cyma. — An  ogee.  There  are  two  kinds  ;  the  cyma-rec.ta,  having 
tht  upper  part  concave  and  the  lower  convex,  and  the  cyma-reversa, 
.with  the  upper  part  convex  and  the  lower  concave. 

Dado. — The  die,  or  part  between  the  base  and  cornice  of  a  pedestal. 

Dairy. — An  apartment  or  building  for  the  preservation  of  milk,  and 
the  manufacture  of  it  into  butter,  cheese,  d:c. 

Dead-shoar. — A  piece  of  timber  or  stone  stood  vertically  in  brick- 
work, to  support  a  superincumbent  weight  until  the  brickwork  which 
is  to  carry  it  has  set  or  become  hard. 

Decastyle. — A  building  having  ten  columns  in  front. 

Dentils. — (From  the  Latin,  denies,  teeth.)  Small  rectangular  blocks 
used  in  the  bed-mouldings  of  some  of  the  orders. 

Dtastyte. — An  intercolumniation  of  three,  or,  as  some  say,  foui 
diameters. 


APPENDIX.  11 

/>«.— That  part  of  a  pedestal  included  between  the  base  and  th« 
?,ornice  ;  it  is  also  called  a  dado. 

Dodecastyle. — A  building  having  twelve  columns  in  front. 

Donjon. — A  massive  tower  within  ancient  castles  to  which  the  gap 
nson  might  retreat  in  case  of  necessity. 

Dooks. — A  Scotch  term  given  to  wooden  bricks.  x 

Dormer.— A  window  placed  on  the  roof  of  a  house,  the  frame  being 
piaced  vertically  on  the  rafters. 

Dormitory. — A  sleeping-room. 

Dovecote. — A  building  for  keeping  tame  pigeons.     A  columbarium. 

Echinus. — The  Grecian  ovolo. 

Elevation. — A  geometrical  projection  drawn  on  a  plane  at  right  an- 
gles  to  the  horizon. 

Entablature. — That  part  of  an  order  which  is  supported  by  the  co- 
lumns ;  consisting  of  the  architrave,  frieze,  and  cornice. 

Eustyle. — An  intercolumniation  of  two  and  a  quarter  diameters. 

Exchange. — A  building  in  which  merchants  and  brokers  meet  to 
transact  business. 

Extrados. — The  exterior  curve  of  an  arch. 

Facade. — The  principal  front  of  any  building. 

Face-mould — The  pattern  for  marking  the  plank,  out  of  which  hand- 
railing  is  to  be  cut  for  stairs,  &c. 

Facia,  or  Fascia. — A  flat  member  like  a  band  or  broad  fillet. 

Falling-mould. — The  mould  applied  to  the  convex,  vertical  surface 
of  the  rail-piece,  in  order  to  form  the  back  and  under  surface  of  the 
rail,  and  finish  the  squaring. 

Festoon. — An  ornament  representing  a  wreath  of  flowers  and  leaves. 

Fillet. — A  narrow  flat  band,  listel,  or  annulet,  used  for  the  separa- 
aon  of  one  moulding  from  another,  and  to  give  breadth  and  firmness 
10  the  edges  of  mouldings. 

Flutes. — Upright  channels  on  the  shafts  of  columns. 

Flyers. — Steps  in  a  flight  of  stairs  that  are  parallel  to  each  other. 

Forum. — In  ancient  architecture,  a  public  market ;  also,  a  place 
where  the  common  courts  were  held,  and  law  pleadings  carried  on. 

Foundry. — A  building  in  which  various  metals  are  cast  into  moulds 
or  shapes. 

Frieze. — That  part  of  an  entablature  included  between  the  archi- 
trave and  the  cornice. 

GaNe. — The  vertical,  triangular  piece  of  wall  at  the  end  of  a  rool, 
from  the  level  of  the  eaves  to  the  summit. 

Gain. — A  recess  made  to  receive  a  t^non  or  tusk. 

Gallery. — A  common  passage  to  several  rooms  in  an  upper  story. 
A  long  room  for  the  reception  of  pictures.  A  platform  raised  on  co- 
lumnr-,  pilasters,  or  piers. 

Girder. — The  principal  beam  in  a  floor  for  supporting  the  binding 
and  other  joists,  whereby  the  bearing  or  length  is  lessened. 

Glyph.— A  vertical,  sunken  channel.  From  their  number,  those  in 
the  Doric  order  are  called  triglyphs. 


12  APPENDIX. 

Granary. — A  building  for  storing  grain,  especially  that  intended  to 
be  kept  for  a  considerable  time. 

Groin. — The  line  formed  by  the  intersection  of  two  arches,  which 
cross  each  other  at  any  angle. 

GultcK. — The  small  cylindrical  pendent  ornaments,  otherwise  called 
drops,  used  in  the  Doric  order  under  the  triglyphs,  and  also  pendent 
from  the  mutuli  of  the  cornice. 

Gymnasium. — Orig'  aally,  a  space  measured  out  and  covered  with 
sand  for  the  exercise  <  f  athletic  games  *  afterwards,  spacious  buildings 
devoted  to  the  mental  as  well  as  corporeal  instruction  of  youth. 

Hall. — The  first  large  apartment  on  entering  a  house.  The  public 
room  of  a  corporate  body.  A  manor-house. 

Ham. — A  house  or  dwelling-place.  A  street  or  village  :  hence 
Notting/m?»,  Bucking/jam,  &c.  Hamlet,  the  diminutive  of  ham,  is  a 
small  street  er  village. 

Helix. — The  small  volute,  or  twist,  under  the  abacus  in  the  Corin- 
thian capital. 

Hem. — The  projecting  spiral  fillet  of  the  Ionic  capital. 

Hexastyle. — A  building  having  six  columns  in  front. 

Hip-rafter. — A  piece  of  timber  placed  at  the  angle  made  by  two  ad- 
jacent inclined  roofs. 

Homestall. — A  mansion-house,  or  seat  in  the  country. 

Hotel,  or  Hostel. — A  large  inn  or  place  of  public  entertainment.  A 
large  house  or  palace. 

Hot-house. — A  glass  building  used  in  gardening. 

Hovel. — An  open  shed. 

Hut. — A  small  cottage  or  hovel  generally  constructed  of  earthy 
materials,  as  strong  loamy  clay,  &c. 

Impost. — The  capital  of  a  pier  or  pilaster  which  supports  an  arch. 
Intaglio. — Sculpture  in  which  the  subject   is  hollowed  out,  so  that 
the  impression  from  it  presents  the  appearance  of  a  bas-relief. 
Intercolumniaiion. — The  distance  between  two  columns. 
Intrados. — The  interior  and  lower  curve  of  an  arch. 

Jack-rafters. — Rafters  that  fill  in  between  the  principal  rafters  of  a 
roof;  called  also  common-rafters. 

Jail. — A  place  of  legal  confinement. 

Jambs. — The  vertical  sides  of  an  aperture. 

Joggle-piece. — A  post  to  receive  struts. 

Joists. — The  timbers  to  which  the  boards  of  a  floor  or  the  laths  of  a 
ceiling  are  nailed. 

Keep. — The  same  as  donjon,  which  see. 
Key-stone. — The  highest  central  stone  of  an  arch. 
Kiln. — A  building  for  the  accumulation  and  retention  of  heat,  in  ci- 
der to  dry  or  burn  certain  materials  deposited  within  it. 
King-post. — The  centre.-post  in  a  trussed  roof. 
Knee, — A  convex  bend  in  the  back  of  a  hand-rail.     See  Ramp. 


APPENDIX.  13 

Lactarium. — The  same  as  dairy,  which  see. 

Lantern. — A  cupola  having  windows  in  the  Bides  for  lighting  an 
apartment  beneath. 

Larmier. — The  same  as  corona,  which  see. 

Lattice. — A  reticulated  window  for  the  admission  of  air,  rather  than 
fight,  as  in  dairies  and  cellars. 

Lever-boards. — Blind-slats  :  a  set  of  boards  so  fastened  that  they 
may  be  turned  at  any  angle  to  admit  more  or  less  light,  or  to  lap  upon 
^ach  other  so  as  to  exclude  all  air  or  light  through  apertures. 

Lintel. — A  piece  of  timber  or  stone  placed  horizontally  over  a  door, 
window,  or  other  opening. 

ListeL — The  same  as  fillet,  which  see. 

Lobby. — An  enclosed  space,  or  passage,  communicating  with  the 
principal  room  or  rooms  of  a  house. 

Lodge. — A  small  house  near  and  subordinate  to  the  mansion.  A 
cottage  placed  at  the  gate  of  the  road  leading  to  a  mansion. 

Loop. — A  small  narrow  window.  Loophole  is  a  term  applied  to  the 
vertical  series  of  doors  in  a  warehouse,  through  which  goods  are  de- 
livered by  means  of  a  crane. 

Lu/er-board/ng. — The  same  as  lever-boards,  which  see. 

Luthern. — The  same  as  dormer,  which  see. 

Mausoleum. — A  sepulchral  building — so  called  from  a  very  cele- 
brated one  erected  to  the  memory  of  Mausolus,  king  of  Caria,  by  his 
wife  Artemisia. 

Metopa. — The  square  space  in  the  frieze  between  the  triglyphs  of 
the  Doric  order. 

Mezzanine. — A  story  of  small  height  introduced  between  two  of 
greater  height. 

Minaret. — A  slender,  lofty  turret  having  projecting  balconies,  com- 
mon in  Mohammedan  countries. 

Minster. — A  church  to  which  an  ecclesiastical  fraternity  has  been 
or  is  attached. 

Moat. — An  excavated  reservoir  of  water,  surrounding  a  house,  cas 
tie  or  town. 

Modillion. — A  projection  under  the  corona  of  the  richer  orders,  re 
sembling  a  bracket. 

Module. — The  semi-diameter  of  a  column,  used  by  the  architect  as 
a  measure  by  which  to  proportion  the  parts  of  an  order. 

Monastery. — A  building  or  buildings  appropriated  to  the  reception  of 
monks. 

Monopieron. — A  circular  collonade  supporting  a  dome  without  an 
enclosing  wall. 

Mosaic.— A.  mode  of  representing  objects  by  the  inlaying  of  small 
cubes  of  glass,  stone,  marble,  shells,  &c. 

Mosque. — A  Mohammedan  temple,  or  place  of  worship. 

Mullions.—The  upright  posts  or  bars,  which  divide  the  lights  in  a 
Gothic  window. 

Muniment-house. — A  strong,  fire-proof  apartment  for  the  keeping 
and  preservation  of  evidences,  charters,  seals,  &c.,  called  muniments. 


14  APPENDIX. 

Museum. — A  repository  of  natural,  scientific  and  literary,  curiosities 
or  of  works  of  art. 

Mutule. — A  projecting  ornament  of  the  Doric  cornice  supposed  tc 
represent  the  ends  of  rafters. 

Nave. — The  main  body  of  a  Gothic  church. 
Newel. — A  post  at  the  starting  o-  landing  of  a  flight  of  stairs. 
Niche. — A  cavity  or  hollow  place  in  a  wall  for  the  reception  of  a 
ftatue,  vase,  &c. 

Nogs. — Wooden  bricks. 

Nosing. — The  rounded  and  projecting  edge  of  a  step  in  stairs. 

Nunnerv. — A  building  or  buildings  appropriated  for  the  reception  of 


Obelisk. — A  lofty  pillar  of  a  rectangular  form. 

Octastyle. — A  building  with  eight  columns  in  front. 

Odeum. — Among  the  Greeks,  a  species  of  theatre  wherein  the  poets 
and  musicians  rehearsed  their  compositions  previous  to  the  public  pro- 
duction of  them. 

Ogee. — See  Cyma. 

Orangery. — A  gallery  or 'building  in  a  garden  or  parterre  fronting 
the  south. 

Oriel-window. — A  large  bay  or  recessed  window  in  a  hall,  chapel,  or 
other  apartment. 

Ovolo. — A  convex  projecting  moulding  whose  profile  is  the  quad- 
rant of  a  circle. 

Pagoda. — A  temple  or  place  of  worship  in  India. 

Palisade. — A  fence  of  pales  or  stakes  driven  into  the  ground. 

Parapet. — A  small  wall  of  any  material  for  protection  on  the  sides 
of  bridges,  quays,  or  high  buildings. 

Pavilion. — A  turret  or  small  building  generally  insulated  and  com- 
prised under  a  single  roof. 

Pedestal. — A  square  foundation  used  to  elevate  and  sustain  a  co- 
lumn, statue,  &c. 

Pediment. — The  triangular  crowning  part  of  a  portico  or  aperture 
which  terminates  vertically  the  sloping  parts  of  the  roof:  this,  in 
Gothic  architecture,  is  called  a  gable. 

Penitentiary. — A  prison  for  the  confinement  of  criminals  whose 
crimes  are  not  of  a  very  heinous  nature. 

Piazza. — A  square,  open  space  surrounded  by  buildings.  This 
term  is  often  improperly  used  to  denote  a  portico. 

Pier. — A  rectangular  pillar  without  any  regular  base  or  capital. 
The  uprighl,  narrow  portions  of  walls  between  doors  and  windows  are 
known  by  this  term. 

Pilaster. — A  square  pillar,  sometimes  insulated,  but  more  common 
ly  engaged  in  a  wall,  and  projecting  only  a  part  of  its  thickness. 

Piles. — Large  timbers  driven  into  the  ground  to  make  a  secure 
foundation  in  marshy  pla  jes,  or  in  the  bed  of  a  river. 

Pillar. — A  column  of  irregular  form,  always  disengaged,  and  al< 


APPEND' X.  15 

ways  deviating  from  the  proportions  of  the  orders  ;  whence  the  distinc. 
tion  between  a  pillar  and  a  column. 

Pinnacle. — A  small  spire  used  to  ornament  Gothic  buildings. 

Planceer. — The  same  as  soffit,  which  see. 

P/inl.'i  — The  lower  square  member  of  the  base  of  a  column,  pedes- 
tal, or  wall. 

Porch. — An  exterior  appendage  to  a  building,  forming  a  covered 
app)oa:Vn  to  one  of  its  principal  doorways. 

Portal.—  The  arch  over  a  door  or  gate  ;  the  framework  of  the  gate  ; 
the  lesser  gate,  when  there  are  two  of  different  dimensions  at  one  en- 
trance. 

Portcnllis. — A  strong  timber  gate  to  old  castles,  made  to  slide  up 
and  down  vertically. 

Portico. — A  colonnade  supporting  a  shelter  over  a  walk,  or  ambu- 
latory. 

Priory. — A  building  similar  in  its  constitution  to  a  monastery  or 
abbey,  the  head  whereof  was  called  a  prior  or  prioress. 

Prism. — A  solid  bounded  on  the  sides  by  parallelograms,  and  on  the 
ends  by  polygonal  figures  in  parallel  planes. 

Prostyle. — A  building  with  columns  in  front  only. 

Purlines. — Those  pieces  of  timber  which  lie  under  and  at  right  an- 
^les  to  the  rafters  to  prevent  them  from  sinking. 

Pycnoslyle. — An  intercolumniation  of  one  and  a  half  diameters. 

Pyramid. — A  solid  body  standing  on  a  square,  triangular  or  poly- 
gonal basis,  and  terminating  in  a  point  at  the  top. 

Quarry. — A  place  whence  stones  and  slates  are  procured. 

Quay. — (Pronounced,  key.}  A  bank  formed  towards  the  sea  or  on 
the  side  of  a  river  for  free  passage,  or  for  the  purpose  of  unloading 
merchandise. 

Quoin. — An  external  angle.     See  Rustic  quoins. 

Rabbet,  or  Relate. — A  groove  or  channel  in  the  edge  of  a  board. 

Ramp. — A  concave  bend  in  the  back  of  a  hand-rail. 

Rampant  arch. — One  having  abutments  of  different  heights. 

Regula. — The  band  below  the  tcenia  in  the  Doric  order. 

Riser. -^- In  stairs,  the  vertical  board  forming  the  front  of  a  step. 

Rostrum. — An  elevated  platform  from  which  a  speaker  addresses  aiv 
audience. 

Rotunda. — A  circular  building. 

Rubble-wall. — A  wall  built  of  unhewn  stone. 

Rudenlure. — The  same  as  cable,  which  see. 

Rustic  quoins. — The  stones  placed  on  the  external  angle  of  a  build- 
ing, projecting  beyond  the  face  of  the  wall,  and  having  their  edges 
bevilled. 

Rustic-work. — A  mode  of  building  masonry  wherein  the  faces  of  the 
stones  are  left  rough,  the  sides  only  being  wrought  smooth  where  the 
union  of  the  stones  takes  place. 


16  APPENDIX. 

Salon,  or  Saloon. — A  lofty  and  spacious  apartment  comprehending 
the  height  of  two  stories  with  two  tiers  of  windows. 

Sarcophagus. — A  tomb  or  coffin  made  of  one  sione. 

Scantling. — The  measure  to  which  a  piece  of  timber  is  to  be  or  has 
been  cut. 

Scarfing. — The  joining  of  two  pieces  of  timber  by  bolting  or  nailirg 
transversely  together,  so  that  the  two  appear  but  one. 

Scotia. — The  hollow  moulding  in  the  base  of  a  column,  between  the 
fillets  of  the  tori. 

Scroll. — A  carved  curvilinear  ornament,  somewhat  resembling  in 
profile  the  turnings  of  a  ram's  horn. 

Sepulchre. — A  grave,  tomb,  or  place  of  interment. 

Sewer. — A  drain  or  conduit  for  carrying  off  soil  or  water  from  any 
place. 

Shaft. — The  cylindrical  part  between  the  base  and  the  capital  of  a 
column. 

Shoar. — A  piece  of  timber  placed  in  an  oblique  direction  to  support 
a  building  or  wall. 

Sill. — The  horizontal  piece  of  timber  at  the  bottom  of  framing  ;  the 
timber  or  stone  at  the  bottom  of  doors  and  windows. 

Soffit — The  underside  of  an  architrave,  corona,  &c.  The  underside 
of  the  heads  of  doors,  windows,  &c. 

Summer. — The  lintel  of  a  door  or  window  ;  a  beam  tenoned  into  a 
girder  to  support  the  ends  of  joists  on  both  sides  of  it. 

Systyle. — An  intercolurnniation  of  two  diameters. 

Taenia. — The  fillet  which  separates  the  Doric  frieze  from  the  archi- 
trave. 

Talus. — The  slope  or  inclination  of  a  wall,  among  workmen  called 
battering. 

Terrace. — An  area  raised  before  a  building,  above  the  level  of  the 
ground,  to  serve  as  a  walk. 

Tesselated  pavement. — A  curious  pavement  of  Mosaic  work,  com- 
posed  of  small  square  stones. 

Tetrastyle. — A  building  having  four  columns  in  front. 

Thatch. — A  covering  of  straw  or  reeds  used  on  the  roofs  of  cottages, 
barns,  &c. 

Theatre. — A  building  appropriated  to  the  representation  of  drainage 
spectacles. 

Tile. — A  thin  piece  or  plate  of  baked  clay  or  other  material  used  for 
the  external  covering  of  a  roof. 

Tomb. — A  grave,  or  place  for  the  interment  of  a  human  body,  in- 
cluding also  any  commemorative  monument  raised  over  such  a  place. 

Torus. — A  moulding  of  semi-circular  profile  used  in  the  bases  of 
columns. 
,    Tower. — A  lofty  building  of  several  stories,  round  or  polygonal. 

Transept. — The  transverse  portion  -f  a  cruciform  church. 

Transom. — The  beam  across  a  double-lighted  window;  if  the  win 
dow  have  no  transom,  it  is  called  a  clere-story  window. 


APPENDIX.  17 

Tread. — Tha ,  part  of  a  step  which  is  included  between  the  face  of 
it?  riser  and  that  of  the  riser  above. 

Trellis. — A  reticulated  framing  made  of  thin  bars  of  wood  foi 
screens,  windows,  &c. 

Triglyph. — The  vertical  tablets  in  the  Doric  frieze,  chamfered  on 
•.he  two  vertical  edges,  and  having  two  channels  in  the  middle. 

Tripod.-  -A  table  or  seat  with  three  legs. 

Trochilus. — The  same  as  scotia,  which  see. 

Truss. — An  a:vangement  of  timbers  for  increasing  the  resistance  to 
cross-strains,  consisting  of  a  tie,  two  struts  and  a  suspending-piece. 

Turret. — A  small  tower,  often  crowning  the  angle  of  a  wall,  &c. 

Tusk — A  short  projection  under  a  tenon  to  increase  its  strength. 

Tympanum. — The  naked  face  of  a  pediment,  included  between  the 
•evel  and  the  raking  mouldings. 

Underpinning. — The  wall  under  the  ground-sills  of  a  building. 
University. — An  assemblage  of  colleges  under  the  supervision  of  a 
senate,  &c. 

Vault. — A  concave  arched  ceiling  resting  upon  two  opposite  paral- 
lel walls. 

Venetian-door. — A  door  having  side-lights. 

Venetian-window. — A  window  having  three  separate  apertures. 

Veranda. — An  awning.  An  open  portico  under  the  extended  roof 
of  a  building. 

Vestibule. — An  apartment  which  serves  as  the  medium  of  commu- 
nication to  another  room  or  series  of  rooms. 

Vestry. — An  apartment  in  a  church,  or  attached  to  it,  for  the  pre- 
servation of  the  sacred  vestments  and  utensils. 

Villa. — A  country-house  for  the  residence  of  an  opulent  person. 

Vinery. — A  house  for  the  cultivation  of  vines. 

Volute. — A  spiral  scroll,  which  forms  the  principal  feature  of  the 
ionic  and  the  Composite  capitals. 

Voussoirs. — A  rch-stones 

Wainscoting. — Wooden  lining  of  walls,  generally  in  panels. 

Water-table. — The  stone  covering  to  the  projecting  foundation  oi 
other  walls  of  a  building. 

Well. — The  space  occupied  by  a  flight  of  stairs.  The  space  left 
beyond  the  ends  of  the  steps  is  called  the  well-hoce. 

Wicket. — A  small  door  made  in  a  gate. 

Winders. — In  stairs,  steps  not  parallel  to  each  other. 

Zophorus. — The  same  as  frieze,  which  see. 

Zyttos. — Among  the  ancients,  a  portico  of  unusual  lergth,  common 
1)  appropriated  to  gymnastic  exercises. 


TABLE  OF  SQUARES,  CUBES.  AND  ROOTS. 

(From  Button's  Mathematics.) 


No. 

Square. 

Cube.      1     Sq.Root. 

Cube  Root. 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeKaot. 

rr 

1                      1 

i-ooooooo'  i-oooooo 

68 

4624 

314432 

8-2462113 

4-081655 

2 

4 

8 

1  4142136 

:  1-25J921 

69 

4761 

328509 

8-3066239 

4  101566 

3 

9 

27 

l-732050d 

!   1-442250 

70 

4900 

343000 

8-3666003 

4-121285 

4 

16 

64 

2-0000001 

1-537401 

71 

5041 

357911 

8-4261498 

4-140818 

5 

25 

125 

2-236068U 

;  1-709976 

72 

5184 

373248 

8-4852814 

4-160168 

6 

36 

216 

2-44948J7 

1  1-317121 

73 

5329 

389017 

8-5440037 

4-179333 

7 

49 

343 

2-645751:3 

!  1-912931 

74 

5476 

405224 

8-6023253 

4-198336 

8 

64 

512 

2-82-44271 

f  2-OOOiXJO 

5625 

421375 

8-6602540 

4  217163 

9 

81 

729 

3-0  IOOODC 

1  2-030034 

I- 

.    5776 

433976 

8-7177979 

4  1235324 

10 

100 

10,  lO 

3-1622777 

2-154435 

5929 

456533 

8-7749644 

4-254321 

11 

121 

1331 

33l6i524S 

2-223J30 

78 

6084 

474552 

8-83176D9 

4-272659 

12 

144 

17-43 

3-4641016 

2-2W429 

79 

6241 

493039 

8-8331944 

4-290340 

13 

169 

2197 

3-6,55:513 

2351333 

80 

6400 

512000 

8-9442719 

4-303K69 

14 

196 

2744 

3-7416574 

2-110142 

81 

6561 

531441 

9-(KKHKXM) 

4-3-.6T49 

15 

225 

3375 

3-8729833 

2-466212 

82 

6724 

5513:58 

9-0553851 

4-34-4481 

16 

256 

40J6 

4-00,10000 

2-519342 

83 

6339         571787 

9-1104336 

4-362071 

17 

289 

4913 

4-1231056 

2-571232 

84 

7056         592704 

9-1651514 

4-379519 

18 

324 

5332 

4-2426407 

2-620741 

85 

72251        614125 

9-2195445 

4-396830 

19 

361 

635  J 

4-353  «39 

2-66  -i402 

86 

7396         63605:5 

9-2736185 

4-414005 

20 

400 

8000 

4-472  13:50 

•^•714118 

87 

7569 

653503 

93273791 

4-431048 

21 

22 

441 

484 

9261 
10643 

4-5825757 
4-69J1153 

2-758924 
2-3G203J 

8S 
89 

7744 
7921 

681472 
704969 

9  3308315'  4-447960 
9-4339311    4-464745 

23 

529 

12167 

4-7953315 

2-343367 

90 

8100 

729!  XX) 

9-4363331) 

4-4B1405 

24 

576 

13324 

4-898.1795 

2-384499 

91 

8281 

753571 

9-5393920 

4-497941 

25 

625 

15625 

5-OJOOOOO 

2-924018 

92 

8164         773683 

9-5916630 

4514357 

26 

676 

17576 

5-0990195 

2-962496 

93 

8649         804357 

96436513 

4  -53/  >655 

27 

729 

19683 

5-1961524 

3-001XJOO 

94 

8836|        8305S4 

9-6953597 

4  516336 

23 

784' 

21952 

5  2915,12(5 

3U36539 

95 

9025         857375     9-7467943 

4-5fi2903 

29 

841 

24389 

53351643 

3-072317 

96 

9216!        834736     9-7979590 

4-573357 

30 

900 

27000 

5-4772256 

3-107232 

97 

9409         9126731    9-8483578 

4-59i701 

31 

961 

'2"  9791 

55(577644 

3  1413J1 

98 

9604 

941192 

9-8994949 

4-610436 

32 

1024 

32768 

5-6568542 

3-174302 

99 

9301         970299 

9-9498744 

4-626065 

33 

1089 

35937 

5-7445626 

3-207531 

100 

10000 

1000000 

100000000 

4  641589 

34 

1156 

39304 

5-83.)9519 

3-23J612 

101 

10201 

103)301 

10-049375:-, 

4657009 

35 

1225 

42875 

5-9160798 

3271066 

102 

10404 

1061208 

10-0995049 

4  '.572329 

36 

1296 

46656 

6-OOOJOOO 

33J1927 

103 

10609 

1092727 

10-1483910 

4-637548 

37 

1369 

50653 

6-0327625 

3332222 

104 

10816 

1124864 

10-1980390 

4-702669 

1  33 

1444 

54872 

6  1644140 

3361975 

105 

110-25 

1157625 

10-246.)5^ 

4-7  '7694 

3J 

1521 

59319 

6-2449980 

3391211 

106 

11236 

1191016 

10-2*155301 

4-732623 

40 

1600 

640.K) 

6-3245553 

3419952 

107 

11449 

1225043 

10-31403U4 

4  7  47459 

41 

1681 

68921 

6-4031242 

3448217 

108 

11664 

1259712 

10-39233W 

4-762203 

*2 

1764 

74038 

6-4807407 

3476027 

109 

11881 

1295029 

10-4403.J65 

4-776856 

*3 

1819 

79507 

6-5574335 

3503393 

-110 

12100 

1331000 

10-4880835 

4-791420 

44 

1936 

85184 

66332496 

353J348 

111 

12321 

1367631 

10-535653S1 

4-805395 

45 

2025 

91125 

6-7082039 

3556893 

112 

12544 

1404928 

10-583005* 

4-820284 

46 

2116 

97336 

6-7823300 

3-583J48 

113 

12769 

1442897 

10-6301453 

4-834588 

47 

2209 

103S23 

6-8556546 

3-608826 

114 

12996 

1481544 

10-67707-33 

4-848308 

43 

2304 

1  10592 

69232032 

3631241 

115 

13225       1520875 

10  7238053 

4-862944 

49 

2401 

117649 

7-OOOOOJO 

3-6593J6 

116 

13456       1560896 

10-77032.^ 

4-87(5999 

50 

2500 

125000 

7-0710678 

3634031 

117 

136891       1601613 

10-816i553.S 

4-890973 

51 

2601 

132*551 

7-1414284 

3-70343J 

118 

13924 

1643032 

10-8627805 

4-904863 

52 

2704 

140608 

7-2111026 

3-732511 

119 

14161 

1685159 

10-9037121, 

4-918685 

53 

2809 

148377 

7-2301099 

3-756286 

120 

14400       1728000 

10-954  i5  I'- 

4-932424 

51 

2916 

157464 

7-3184692 

3-779763 

121 

14641       1771561 

ll  -tXMXKKJii 

4-946387 

55 

3025 

166375 

7-4161935 

3-^02952 

122 

14884       1815848 

ll-0153n!i; 

4-959676 

56 

3136 

175616 

7-4833148 

3-825862 

123 

15129       1860867 

11-09053.55 

4-973190 

57 

3249 

185193 

7-5493344 

3-843501 

124 

15376       1906624 

11-1355237 

4-936631 

53 

33(54 

195112 

7-6157731 

3-870377 

125 

15625        1953125    11-KW33.W 

5-000000 

59 

3431 

205379 

7-6311457 

3-392996; 

126      15876       2000376,   11-2219722 

5-013298 

60 

3600 

216000 

7-7459667 

3-9148M! 

127      16129       2048333    11-2694277 

5-026526 

61 
62 

3721 
3t44 

226931 
233323 

7-8102497    3-<J3<54J7 
7-8740J791  3-957391 

I28l     16334       2097152)   11-3137035J  5-039631 
129      166411       2146639!    ll-35V«*U??l  5-052774 

6.< 

3J6J 

250047 

7-9372539 

3-9790571 

130 

16JOO'       2197000;    11-4017543 

5.165797 

6| 

4'iyrt 

262144 

8  -0000,  MO 

4-00'JOOOl 

131      17161       2248091!  ll-4455s8:U 

5'07t<753 

f>') 

4225 

274025 

8-0622577 

4-02J726! 

132 

17-124!       229996.-!    11-13.11^5^ 

5-091643 

6ii 

67 

•tiVi 

4l3.t 

2374J6 
300763 

8-  1240334 
8-1853523 

4-041240! 
4-0615431 

133      17689       235*637    I1-5325.52S 
134      17956       24.x;  KM  ;  11-^753369 

5  104469 
5  117230 

APPENDIX. 


10 


\0. 

Square.         Cube. 

S<j.  Root. 

CubeRoot. 

No. 

Square. 

Cube.      I     Sq.  Root. 

ICubcRnol. 

133 

18225 

2460375 

11-6189500 

5-129928 

202 

4ii804|       8242408'  14-212fi7()4l  5S674fi* 

136 

18496 

2515456 

ir-6619038 

5-142563 

203 

412091       83(55427 

i  14-247SOI5' 

il  5-877131 

137 

18769 

2571353 

11-7046999 

5-155137 

20-4 

4I616J      848966-J 

14-2328561 

>l  5-836765 

13S 

19044 

2628072 

11-7473401 

5-167649 

205 

42025 

8615123 

•  14-317321 

5-896366 

139 

19321 

2685619 

11-7898261 

5-180101 

206 

42  4* 

87418K 

i  14-352700 

5-905J41 

140 

19600 

2744000 

11-8321596 

5-192494 

207 

42849 

RS6974L 

14-387494f 

1  5-915482 

141 

19881 

2803221 

11-3743422 

5-204828 

203 

43264 

8998912 

1  11-422205 

!  5-924992 

142 

20164 

2863283 

11-9163753 

5-217103 

209 

43681 

912932ii 

14-456832I 

ti  5-934473 

143 

20449 

2924207 

11-9582607 

5-229321 

210 

44100 

9261000 

14-491376' 

I  5-943.122 

1441    20736 

2985984 

12-0000000 

5-241483 

211 

44521 

9393931 

1  14-5258391 

1  5-953342 

145     21025 

3048625 

12-0415946 

5-253538 

212 

44944 

9528128 

!  14-560219; 

i  5-9(52732 

146 

21316 

3112136 

12-0830460 

5-265637 

213 

45369 

9663597 

14-5945193 

!  5-972093 

147 

21609 

3176523 

12-1243557 

5-277632 

214 

45796 

9300344 

14-628733; 

i  5-931421 

148 

21904 

3241792 

12-1655251 

5-289572 

215 

46225 

9933375 

14-662378J 

!  5  9b072j 

149 

22201 

3307949 

12-2065555 

5301459 

216 

46656 

10077(596 

14-696933S 

i  6-ouooo,/ 

150 

22500 

3375000 

12-2474437 

5-313203 

217 

47089 

10218313 

14-730919S 

!  6-00  J2  4  5 

151 

22801 

3442951 

12-2332;)57 

5-325074 

218 

47524 

10.360232 

14-7648231 

6-018462 

152 

23104 

3511808 

12-3288280 

5-336803 

219 

47961 

10503459 

14-798S48G 

6-027650 

153 

23409 

3531577 

12-3693169 

5-348431  !  220 

48400 

10643000 

14-8323971 

6-036311 

154 

23716 

3652264 

12-4096736 

5-360103    221 

43341 

10793361 

14-8660(587 

!  6-045943 

155 

24025 

3723375 

12-449399.. 

5-371635  !  222 

49234 

10941048 

14-89J6644 

;  6-055049 

156 

24336 

3796416   12-4899960 

5-333213 

223 

49729 

11039567 

14-9331843 

6-0641-27 

157 

•24649 

3869393    12-5299641 

5-394691 

224 

50176 

11239421 

14-9656295 

!  6-07317^ 

153     21%  4 

3944312 

12-5698051 

5-406120 

225 

50625 

11390625 

15-0000000 

i  6-^2202 

159|     25281 

4019679 

12-6095202 

5-417501 

226 

51076 

11543176 

15-0332961 

;  G-Oyil'JsJ 

160     2'>600 
In  I      2f)921 

4096000 
4173281 

12-6491106  5-428335 
12-6335775   5-440122 

227 
223 

51529 
51984 

11697083 
11352352 

15-0665192   6-100170 
15-099663-J;  G-1-.W115 

4251528 

12-7279221)  5-451362 

229 

52441 

12008939 

15-1327460 

6-113033 

163!    265(59 

4330747 

12-7671453  5-462556 

230 

52900 

12167000 

15-1657509 

6-126J2j 

164!    £6896 

,  4410944 

12-8062483  5-473704 

231 

53361 

12326391 

15-1986842 

6-1357^2 

1(55     V7225 

4492125 

12-84523261  5-4313:>7    232 

53324 

12487168 

15-2315462 

6-144(534 

16-5'     t7556 

4574296    12-88409871  5-495365    233[    54239 

12649337 

15-2643375 

6-153449 

16?     27839 

4657463 

12  9228480  5-506378 

234 

5475:. 

12812904 

15-2970585 

6-  1622  W 

163     £$224 

4741632    12-9614814]  5-517348 

235 

55225!     12977875 

153297097 

6-171005 

169     2S561 

4826809   13-0000000 

5-523775 

236     55696!     13144256 

159622915 

6-179747 

170     -iSJOO 

49130001  13-0384048 

5-539653 

237 

56169      13312053 

15-3J43043 

6-183463 

171     29241 

5000211    13-0766963 

5-550499 

•233 

56644     13481272 

15-4272486 

6-197154 

172     29'>84 

5033448:  13-1143770 

5-561298    23J     5712l!     13651919 

15-4596243 

6-205322 

173 

29929 

51777171  13-1529461 

5572055.    240     57600      13324000 

15-4919334 

6-214465 

rw 

30276 

5-268024!  13-1909060 

5532770    241!     58031      139..7521 

15-5241747 

6  22303  i 

175 

30625 

53593751  13-2287566 

5-593445    242'     53564!     H172433 

155563192 

6-2316*) 

176 

SfKPG 

5451776    13-2*564992   5-604079!l  243     5904lj|     14343907 

15-5334573 

6-240251 

177 

31329 

5545233    13-3041347   5-614672    244     59536 

14526734 

15-6204994 

6  243*0  j 

178 

31(584 

5639752    13-3416641 

5-625226    245)     60025 

147061-25 

15-6524753 

6-25732-) 

179 

32041 

5735339    13-37-JOS>2 

3-635741     2461     60.')  16 

14336936 

15-6343371 

6-2'55327 

180 

32400 

5832000J  13-4164079J  5-646216    217;     61U09 

15069223 

15-7162333 

6-274a>J 

181 

32761 

59297411  13-45362401  5-656(553    248     61504 

152529J2 

15-7480157 

6-232761 

182 

33124       60285681  13-4907376;  5-667051JJ  249!    62001 

15433249 

15-7797333 

6-291195 

183 

334391      6128-1871  13-5277493   5-677411!    250:    62500 

15625000 

15-8113383 

6-299605 

184 

338361       6229504    13'564(5600i  5-637734    251  1     63001 

15313251 

15-8429795 

63J7^^4 

135 

34225       63316251  136014705   5-698019!    2521     63504 

16003008 

15-8745079 

6316360 

[186 

1  187 

34596 
34969 

6434356!  13-6331817 
6539203;  13-6747943 

5-7os2i57i   253     64009 
5-718479    254     64516 

16194277 
16387064 

15-9059737i  6-32  !70i 
15-9373775   6-m>2'. 

183 

35344 

6644672    13-7113092   5-728i>54i   255     65025 

16581375 

15  9687194 

6-341  32i 

189 

35721 

6751269    13-7477271!  5-738794    256     65536 

16777216 

16  0000000 

6  34915  '4 

190i    36100 

6859000    13-7840483!  5-7483J7J    2571     66049 

16974593 

160312195 

6-357861 

191     36481 

6967871 

13-8202750'  5-758965:    253     66564 

17173512 

16-0623734 

192i    36864 

7077888    13-85640651  5-768998JI  259|    67081 

17373979 

16-0934769 

6371311 

1931    37:249 

7189057   13-8924440!  5-773J96    260!    67500 

17576000 

16-1245155 

194     37<53G 

7301384 

13-9283883]  5-783960 

261     63121 

17779531 

16-  1  554  J  44 

195     38025 

7414875 

13-9G42400   5-798390 

2621    6-644 

17984723 

15-1864141 

6-39332:1 

196i    38416 

197|    38809 
198     39204 
199     39601 

7529536    14-0000000!  5-808736 
7645373   14-0356633   5-818648 
77623921  14-071-2473:  5-32-1477 
78805991  14-10673(50   5-833272 

263!    69169 
264'    6J696 
2t55     70225 
266     70756 

HUH  117 
13399744 
13609625 
1382109,5 

16-2172747 

16-2738206 
16-3095064 

6-4%J53 
6-415'JSa 

6:42315 
6-13122- 

200     40000 

8000000'  14-14-,'135(;l  5-818035 

267     71289 

19034163 

16-340134(5 

6-439277 

201 

40401       8120601)  14-1774469'  5-857766 

263     71824,     1924S33J 

16-3707055 

6-44730t5 

53 


20 


APPENDIX 


Na 

Square. 

Cube. 

Sq.  Root. 

CubeRoot..  1  No.  |  Square. 

Cube. 

Sq  Root.  CubeRoot-  . 

289 

72361 

19465109 

16-40121951  6-455315 

336 

1  12896 

37933053  18-3303028  6952053 

•270 

74900 

19633000 

16-43167671  6-463301 

337 

113569 

38272753 

183575593 

6-958913 

271 

73141 

19902511 

16-4620776  6-471274 

333 

114244 

33614472 

18-3347763 

6-965820 

73984 

20123648 

16-49242251  6-479224 

339 

114921 

38908219 

18-4119526 

6-972683 

273 

74529 

20346417 

16-5227116 

6-487151 

340 

115500 

393"  1000 

18-4390839 

6979532 

274 

75076 

20570824 

16-5529454 

6-495065 

341 

116281 

39651821 

184661853 

6-986363 

*-27r) 

75025 

20796375 

16-5331240 

6-5)2957 

342 

116964 

40001638 

13-4932420 

6-993191 

276 

76176 

21024576 

16-6132477 

6-510830 

343 

117649 

40353607 

18-5202592 

7-000000 

277 

76729  21253933 

16-6433170 

6-518634 

344 

1  18336 

40707534 

18-5472370 

7-006796 

278 

772311  21484'J52 

1667333-20 

6-526519 

345 

119025 

41063625 

18-5741756 

7-013579 

'27^ 

77*11 

2171763-3 

16-7032931 

6-534335 

346 

119716 

41421736 

18-6010752 

7-020349 

•240 

78400 

21952000 

16-7332005 

6-542133 

347 

120409 

41781923 

18-6279360 

7-027106 

231 

73961 

22188041 

16-7630546 

6-549912 

34  3 

121104 

42144192 

186547581 

7-033350 

282 

79524 

22425763 

16-7923556 

6-557672 

349 

121801 

42508549 

18-6315417 

7-040531 

233 

80089 

22665187 

16-8226033 

6-565414 

350 

12-2500 

42875000 

18-7032869 

7-047299 

234 

80656 

22906304 

16-8522995 

6-573139' 

351 

123201 

43243551 

18-7349940 

7-054004 

235 

81225 

23149125 

16-8319430 

6-530344!  i  352 

123904 

436142(8 

187616531 

7-060697 

•28(5 

81796 

23393656 

169115345 

6543532  353 

124609 

4398OT7 

18-7332<)!2  7-067377 

237 

8-2369 

23639903 

16-9410743 

6-596202 

354 

125316 

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18-8148377  7-074044 

283 

82944 

23337872 

16-9705627 

6-603354 

355 

126025 

44733875  18-8414437!  7-030699 

289 

83521  24137559 

17-0000000  6-611489 

356 

126736 

45118016  13-8679623  7-037341 

290 

84100  24389000  17-0293364  6-619106 

357  127449 

45499293,  13-3914136  7-093)71 

291 

84681  24642171  17-0537221  6-626705 

353|  128164 

45882712  18-92  8379"  7-  100533 

893 

852641  24897083'  17-0880075  6-631237 

359 

128331 

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13-9472953  7-107194 

293 

85349  251*3757  17-1172423  6641852 

360 

129600 

46651000 

18-9736660 

7-113737 

294 

86136 

25412184 

17-1464232  6-619400 

36! 

130321 

470?5!81  19-0000000 

7-120367 

295 

87025 

25672375!  17-1755610  6-656930 

352 

131044 

47437928  19-0262976 

7-126936 

296 

87616 

25931336  17-2046505  6-664444 

363 

131769 

47832147 

19-0525539 

7-133492 

297 

8S299 

25198073  17-2336379 

6-671940 

354 

13249^ 

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19-0737840 

7-140037 

298 

8S804 

26463592  17-2626765 

6-67942') 

365 

133-225 

43627125  19-1049732 

7-146569 

299 

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26730899  17-2916165 

6-636383 

366 

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49027395  19-1311265 

7153090 

3.K) 

90000 

27000000  17-32!)5'M1 

6-69432J 

367 

134639 

49130863  19-1572441 

7-159599 

301 

90691  272709011  17-3493516 

6-701759 

368 

135424 

4983503-2!  1'.)-  1833251 

7-166096 

302 

91204  27543603  17-3781472 

6-709173 

369 

136161 

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19-2093727 

7-172531 

303 

91809  27818127!  17-4063952 

6-716570 

370 

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19-2353341 

7-179054 

304 

92416!  28094464!  17-4355953 

6-723951 

371 

137641 

51064811 

19-2613603 

7-185516 

305 

93025  28372625  17-4642492 

6-731316 

372 

138381 

51478848 

19-2373J15 

7-191966 

336 

93636  23652616  17-4928557 

6-733664 

373 

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51895117 

19-3132079 

7-198405 

307 

94249 

23934443  17-5214155 

6-745997 

374 

139876 

52313624 

19-3390796 

7-204832 

308 

94864 

29218112  17-5I992S3 

6-753313 

375 

140625 

52734375  19-3649167 

7-21K48 

95481 

29503529  175783953 

6-760614 

376 

1U376 

53157376'  19-3907194 

7-217652 

96100 

29791000!  17-6068169 

6-767899 

37" 

142129 

535826331  19-4164878 

7-224045 

„.. 

96721 

300302311  17-6351921 

6-775169 

378 

142334 

54010152  19-4122221 

7-23*427 

31-2 

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30371328'  17-6635217 

6-782423 

379 

143641 

54439939  19-4679223 

7-236797 

313 

97.169  3:)664297j  17-6918060 

6-789661 

330 

144400 

54872000 

19-4935337 

7-243156 

311 

985961  30959144J  17-7200151  6-796834 

331 

145161 

55306341 

19-5192213 

7-249504 

315 

99225 

31255375  17-7432393  6-804092 

332 

145924 

55742963  19-5443203 

7-255341 

31 

99S5f 

31554496  17-7763383!  6-811235 

:K? 

1466-19  56131887  19-5703353 

7-262167 

31? 

;  100489 

31855013!  17-8044933!  6-813162  331 

147456  56623104  19-5959179 

7-268432 

318 

101124 

32157432;  17-8325515'  6-825624 

335 

143225;  57066625  19-6214169 

7-274736 

3  IS 

•  101761 

32461759  17-8605711  6-832771 

336 

118996  57512456  19-6468327 

7-231079 

32L 

i  102400 

327630001  17-8835133  6-833904 

337 

149769'  57960603  196723156 

7-287362 

32] 

!  103041 

33076161]  17-91647291  6-847021 

333 

150544!  53411072  19-6977156 

7-293633 

32-J 

:  103(384 

33336213  17-9  1  1358  4  j  6-854124 

339 

151321  53363869  19-7230829 

7-299894 

32: 

!  104329 

33698267  17-9722)08  6-861212 

390 

152100:  59319000J  19-7434177 

7-306144 

324 

104976 

34012224  18-00000!X)i  6-868285 

391 

152831!  59776471  19-7737199 

7312333 

325  105625 
326;  106276 

34323125  18-0277564  6-875344 
34645976  18-0354701  6-832339 

392 
393 

153664  60236233;  19-7939399 
154449  60693457  19-3242276 

7-318611 
7-324829 

32' 

10A98S 

34965783  180331413  6-839419 

391 

155236  61162934  19-84943321  7331037 

i  32; 

!  10758) 

35287552  18-1107703  6-896435 

395 

lc>6025  61629875  19  8746069J  7-337234 

3^1 

1  108241 

35611239  18-1333571 

6903136 

396 

15'816  62099133  19-8997437  7-343420 

33 

10890C 

35937000  18-1659021 

6-910423 

397 

157609;  62570773  19-9243533!  7  349597 

33 

1  109561 

3626469  li  18-1934054 

6-917396 

3.»8 

158404!  630447921  19-9499373 

7-355762 

1  33; 

1  110224 

36591368!  18-2208672  6-924356 

399 

159201 

63521199  19-97  4  0,3  11 

7-361913 

1  33; 

»  110831 

36926037  18-2482376|  6-931301 

400 

160000  64000000!  20-0000000 

7-363063 

:^3 

1  11155f 

37259704  18-2756669!  6-933232 

401 

1608011  64431201  200249344 

7-374198 

335!  11222' 

37595375  18-30300521  6-945150 

402 

161604;  64964808  20^99377 

7-33032 

APPENDIX. 


21 


10-, 


404!  1632 ic,|  £5939*; i 

•105  164025  C643J125  20-121611 

406  164836J  66923116!  20-149441 

407i  165  -49|  67419143i  20- 17421 19!  7-1107951    474    224676!  100496124   2i-771541i:  7-796974 

408,  M6464J  67917312!  20-1990099   7'416859 !  4751  225025!  107171875   21-7914947!  7-802454 

409|  1672S1  6MI7929  2>-223748i!  7-122'.'14     476   220576    107850176   21-8174242!  7-807925 


410    168 100 


411  163921 

412  169744 
413 

171391 
172225 
17305' 
173189 
174721 
175501 
420!  17640U 

421  177241 

422  178U84 
423'  173929 

424  17.1776 

425  180625 
4261  181476 
4271  182329 
4281  183184 
429]  1840H 
43oj  1849;;o 
131     I8--701 
432    185024 
433 


63921000   -20-21845671  7'42895'JJ    477   227529 
69420m    2J-2731349!  7'434994l|  478!  «»484 
699341  28:  20-29778311 
70444997   -20-32-21; U 


7-4410191  479|  229441 
7-447034    480  230400 


2304-JO 
231361 
482'  2323-24 
4-13   2332.19 


108531333  21 -84032  J7j  7-813389 
109215352  21-86321111  7-818846 
109902-239!  21-88606861  7-824294 
1 105920.0;  21-908,023  7-829735 
111284641,1  21-9317122  7-835169 
111980168  21-95 149  vi'  7810595 
H26785i7j  21-97726101  7846013 


7-470999  4S4  '234256  Il33;9j04i  2,>-lHXHX)JO 

7-476966  4<>  235225  1141-84125  ^2-0227155 

7-482921;  4S6  '23611=6  114791256  -2-20454077 

7-488872'  487  237169  115>01303  i-2-068J765 

7-4943111  488:  23S141  116:14272  220^07220 


7-5J0741 
75J6661 


489 


22'H33444 


4.K)   2HH001  117649J(KI   22  135943^i 


7851424 
7-35682 
7-852224 
7-8676 1 : 
7-87WW 
7-87836J 
833735 


7-512571     491    2110S1    118370771    22-15851^;  7-889095 
7-518173    4.12    2i2064    1190J5W8J  i2-l«lo73j!  7-894447 


70957944]  20-3169399 
71173375  20-37 15  1.8s 
71991296  20-390073 
72511713  -29-420577: 
73031632  20-4450483 
73560059  20-46J4895 
740330001  20-4939013 
74618461!  20-5182345 
75151448  20-5420386 
756*5967  2>566.)63S 
70225;  i2i  20-5912603 
70705525  2J-OI55281 

77303776  '20-i',3.»7674  7-524365  493  213049  Il9843i57  22  2U35033  7-899792 
77851,' 3  20-G63J733J  7'530243  4.U  '211035  120553781  w^jllus1  7-905129 
73402752  20-0831609!  7-535122  4J5  245025  121287375  -2^^5.155  7-91046( 
73953589  20-7123152!  7511987  490  24001..  122i-23j36'  22  271u5/fl  7-9157«3 
7;>507000  2073014!!  7'5i73ic  i97  2-iA<09  122763473  22 5JJ34968  7-92109V, 
80062991  20-7605395:  7'55363'J  49s  248004  1235,59.2  a2'315jl3i  7-92640 
80621568  20-7816097  7 -559526  499  24 900 1:  124251499  22  J333J79  7-9317K 
.,  81182737  20-M08652)  7555355  500  25:XXK>  1250,10000  22  3G(X,7^3  7-u370i.5| 

434  183356:    81740504   20-8320667!  7'571174i    501;  251001    125751501    -22  3830293   7-942293 

435  13.1225      ,S2312,i75    20-856652.-5    7-57(5935    502   -25200 11  126506008   224053565   7-947574 

436  190096!    82881856   20-881X5 13)    7-5127.K5    503   253009    127^63527   22'i'2;t;6l5    7-952848 

437  190959.    83153453   20-9045450|  7-583579!    504'  254016    128024064    224499143   7-953114 
433    191814!    84027072    20-92,84495   7-5J43G3    505   255025    12378702')!  22'47'2^051    7-963374 

439  192721!     84601519   20-95232(53.  7000133    50(5   256036    129554216'  -224944433   7-968627 

440  193600]    85131000   20-9761770   7-005905    507    25704 9    130323.813   2251i,05i,5   7-J73373 

411  194481!     85765121    21-0000000   7-0110(53    508   25^01    1310JI.512    :J2538?553   7-97911i 

412  195364      80350338   21-0237900   7-617412     509   '25iHi81    131872^2J|  22oulOi83   7-984314 

443  1962191     86938307   21-0475552   7-023152    510   200KKI    1 32-55 10JO'  22-5831795    7-93957i 

444  197136!     87528334    21-0713075    7-623884.    51  li  261121    133432331,  a2-ou53091    7-994788 

445  19802%    83121125   21-09502311  7-634607    512!  202144    134217723;  ^2-6^74170   80001H)! 
44(3    198916     88716530   21-1187121!  7610321     513   203169    1350056J7J  22-6195033   8-0052( 
447    1998091 

443   2007()4: 


89314623  21-1423745  7'646027  514  264196  135796741:  22-6H5J81  8-01  •4'..-: 

— . ..  89915392  21-1660105  7-051725  515  265225  1365J087.>  -22-6ii36114  8-0155'X 

449;  2016011  90518849  21-18962J1!  7657414  516  21^5256  13738^096  227156334  8-020779 

450  202500:  91125JOO  21-2132034  71563194  517  267289  133l8v413!  22-7376340  8-025957 

268324  138-J91832  22-75Jol3l  8-0311^. 

269351  13979.835J  ^27815715  8  03629:- 


451;  203401J  91733851  21-2367606;  7'668766  51,> 

452  204304  92345408  21-2,<502916I  7'674430  51ljJ , 

453  2J5209i  92959(577  21-2837967    7'C>80086  52O1  '270400  140608000'  228^35085   S-04145 

454  206116^  93576664  21-3072758   7'6,?5733  521  271441  14142,)76l    228,54-44   8-U46603 
45V  207025;  94196375  21'330729o!  7-691372  522  272484  14^236648'  228t;3h.3  8-051748 


456  207936 

457  208819 

458  209764 
45  J  210581 

460  211600 

461  2 1252 1 
4(52  -213411 
463  21436J 


94818816  21-3541555  7  6J7002 

95413993  21  -3775513  7'702(525 

%071912  21-4009346  7'70823J 

96702579  21-4212853  7'713845 

97336000  21-4476106  7'719443 


5'23    '27:s529    143055667J  22v>691933 


521    U7J576    1 


.22-8J10463   806201s 


525    2750-25  144703 U5 

520   '270676  H5531570  22-:'3,0  -99  8-072-62 

527    27772.1  14'>o63183  22-95  ;43.;6  8-077374 

97972181   21-4709106    7'725  )32     523   273734  147197952  22-'J7s^5,K,  8 1*82480 

93  51 1123    21-4941853    7730611     5i9    27'.K11  118035^8.-  '-300.JOJUO  8  O87579 

99252847   21'51743i8   7-73t518.-<    530!  2309JO  1483(7oO,i  U3O.17.8.'  8-09267.; 


464  215296     9J897344   21-5406592    77117; 

465  216225;  1005.14625   21-56385^7    7-7173) 


7-741753  531;  231961  1497212J1  ^3'04:>4372  tt-oy775'j 

I5:  100514625   21-5638587    7'7473ll'  532'  233024  15u55S76>  23-<:65i^5;  8  102-3.' 

466   217150!  101194696   21-5870331    7'75286i  533!  23403J  15141iM.>7  23  08079i8  8-10791^ 

467.  218iW9!  101817503   21-6101823;  7-7534u2  53i|  '^5156  15'227330i  -.31-84400  8-112ir-v 

219024:  1025U3232   21-6333.J77i  7-703.36  535!  286225  153130375  23-13.>U070  8  ll-^oi! 

469i  219961;  10)161709    21-65610781  7-769462  536;  2872U6  153990656  23-1516738  8-123. -J 


APPENDIX. 


No. 

Square. 

Cube.       |     8q.  Root.  . 

CiibeRoot. 

No. 

Square.         Cube. 

8<).  Ro.:t.    jCsteRoou 

537 

288369 

154854153  23  173261)5'  8-12-5145 

6iW 

3.4816 

220348364 

245764115 

8  453-329 

£33 

239444 

155720872   2:H!'43.:7o!  8-1331871 

605 

3  -6025 

221445125 

24-5907478 

8-457691 

539 

2905211   156590319   23-21t>3735|  8-133223: 

600 

3'i7236 

2225450161  24-617C-673 

8-4623  1H 

540  2916001  157464000:  23-2379001!  8-143253 

607 

368449 

•2^<5435-J3   24-6373700 

8-4070-0-J 

541    292631'  153340421    23-2594067J  8-148276! 

608 

369664 

224755712   24-657(5500 

8-471  6  »7 

542 

2937641  159220088!  232-03935;  8-1532941 

009 

3708>:ii  225860529   24-0779254 

8-47628-9 

543 

294349    160103007,  23-3023604J  8-1533051 

010 

372100 

r269810(JO 

24  6981781 

'8-480926 

544 

295936 

160939184   23-323S076   8-163310! 

611 

373321 

22809913! 

247184142 

8-435358 

545 

297025 

161873525   23-3152351J  8-1683J91 

612 

371554 

229220923 

24-7386338 

8490185 

546 

298116 

162771336   23-3666429]  8-173302; 

613 

3  757'  ,9 

230346397 

24-7588368 

8-494806 

547 

299209 

163667323   23-3330311 

8-178233 

6H 

;j7(yj(.;{> 

231475544 

24779IW34 

8-499423 

543 

300304 

164566592   23-4093998 

8-183269 

615 

37,5225 

232608375 

247991935 

8-504035 

549 

301401 

165469  149  !  23-4307490 

8-188244 

616 

379456 

233744895 

24-81934-73 

8-508612 

550 

302500 

1663750001  23-4520788 

8-193213 

617 

330689 

234385113 

24-8394347 

8-513243 

551 

3J3601 

167234151i  23-4733392 

8-198175 

618 

331924 

236029032 

24-3590058 

8-517340 

552 

3!)4704 

163196603!  23-4946802   8-203132 

619 

333161 

237176659 

21-8797106 

8-522432 

553 

3J5809I  16911237.7 

23-5159520]  8-208032 

620 

331400 

233328,;00 

21-3997992 

8-527019 

554 

306916    170031464 

235372046!  8-213027 

6-21 

335641 

239433061 

21-9198716 

8-531001 

555 

303025 

1701)53375   23  5534380!  8-217966 

622 

380334 

240641848 

24-9399278 

8-536178 

556 

309136 

171879610J  23-5790522!  8-222893 

623 

38  129 

241304367 

8-540750 

557 

310249 

172308693!  23-6003174    8-227825 

624 

333370 

242970624 

a4-97999ac 

8-545317 

553 

3113S4 

173741112!  236220236    8-232746 

625 

390&J5 

214140625 

25-0000000 

8-549880 

559 

312431 

174676-I79;  23-6431808!  8-237601 

020 

39187ft 

24531437t 

25-019992  i 

8-551437 

560 

313600 

175:516000   23-6643191 

8-242571 

!  627 

393129 

"246491833 

25-0399681 

8-553990 

561 

314721 

176-')53481    23-0854:!  >6 

8-247474 

••  623 

3943  « 

247673152 

562 

315344 

1775  VISAS'  *''i'7rr>  '>'»')•' 

8-252371 

621 

395641 

248853189 

25-0793724 

8-568W 

563 
564 

316969 
313096 

173453547}  H37276210 
1794:iol44;  237486342 

8-257263 
8-262149 

630 
631 

89690*. 

3J3161 

2500-tTOOO 
2512395J1 

25-099300^ 
25-1197134 

8-5r-26  19 
8577152 

5(55 

319225 

130362  1-25  j  23769728:5 

8-267029 

632 

3J9424 

2524359S8 

25-13'J.'5102 

566 
567 

568 
569 

320356 
321489 
322624 
323761 

1-31321496 
182234263 
133250432 
1842-20009 

237J07515 
23  -ril  170  13 
23-83275;.'6 
23*587209 

8-271904 
8-276773 
8-281635 
8-286493 

;  633 
:  634 
635 
036 

400689 
401956 
403225 
404  196 

253530137 
254810104 
256047375 
257259450 

•25-  159  W!  3  8-5,36205 
•25-17y35r,6  8'59072-t 
25-19921  63   8'595233 
25-2100404  8-5-.W7W 

570 

324900 

185193000 

23-8740728  8-2  J  1344 

637 

405769 

258174853 

'252335539 

8-R042S2 

571 

320011!  136169411 

238.*5!)063'  8-290190 

j  633  407044 

259694072 

25-258S6VJ 

8-6^)875i 

572 

327184    187149248 

23-9165215   8-301030 

639   408321 

200917119 

25  273  1493 

573 

328329    183132517 

83-03741841  8-3J5SG5 

i  6101  4096,  X. 

262144(XK 

25.2932213 

'  8-6  17739 

574   329476|  189119224 

23-9532971 

8-310694 

i  641    410881 

•2ti;i374721 

25-3179778 

tit72-222"- 

575'  330625]  190109375 

23-J791570   8-315517 

!  042j  412164 

204(509283   25'3377189 

576   331776    191102976 
577  332929    19210J033 

24-OOOOOOOJ  8-3203:<5|    643|  413449 
21-02082431  8-325147'   6u!  414736 

265847707!  25  3574447J  8-631183 
207089984   25"3771551    8-035655 

578   334034    193100552 
579!  335241    194104539 

21-0416396   8-329954 
24-0624188!  8-334755 

645!  416025 
616!  417316 

208336125]  '^5  3908502|  8.6-1012S 
269586136!  25'4  1053,11  1  8-644535 

580   3364001  195112000 

24-0331891   8-339551 

!  647   418609 

270340023  25-43.51947 

[  8-649M4 

531    337561    196122941 

24-103J416 

8-344341 

!  6-13   419904 

272097792   25  "45584  41 

8-653497 

582;  333724 

197137368 

24-  124676-2 

8-349126 

1  6491  421201 

2733591491  25-4751781 

1  8-657940 

533!  339339 

193155237 

241453929 

8-353905 

1  6501  42250( 

274625000!  25'495097f 

*  8-602391 

534 

341056 

199176704 

24-1660919 

8-35.3678 

651 

423801 

275894451)  25'5l4701f 

!  8-066331 

535 

342225 

200201625 

24-1867732 

8-363447 

652 

425104 

277167808   25'534290- 

!  8-671266 

586 

34339G 

201230056 

24-2074369 

8-363209 

653  426409 

27,34450771  25'553364* 

1  8675697 

587 

344561) 

202262003 

24-2-230829 

8-372967 

654  427716!  279726204 

25-573423" 

!  8-680124 

588 

345744 

203297472 

242487113  8-377719 

655!  429025!  281011375 

25-592967!! 

;  8-684546 

589 

3469211  204336469 

24-2693222 

8-332465 

656   43>33G 

2823004  If 

25-6124961 

8-683963 

59C 

343100   2J537i>000 

21-2399156   8-337206 

657'  431641) 

26359339C 

25-6320  lh 

8-693376 

591 

349281   -206425071 

24-3104916   8-391942 

[   653!  432964 

234890312   25'651510- 

8-697734 

<SQS 

3504641  207474083 

24-3310501    8-396673 

,    659    1:51231 

286191179   25'(570995: 

;  8-70218S 

591 

351649!  203527857 

24-3515913   8-401398 

\,  660!  43560( 

287496000'  25-690165: 

;  8-700533 

594 

352836  209584584 

24-3721152  8-406116 

1  661;  436921 

288304781;  25709920: 

!  8-710983 

59: 
59f 

3540251  210644875 
355216!  211708736 

24-3926218   8-410332 
24-4131112  8-415542 

B   662;  433244 
J   663   43956'j 

290117528;  2572936071  8-715373 
291434247   257487804   8-719760 

59' 

356409'  212776173 

24-4335834   8-42024f 

664  44089(5   292754944   25768197: 

8724141 

59J 

3576041  213347192 

24-4540385!  8-424945 

:  665   442225   294079625   257875931 

!  8728518 

591 

358801   214921799 

24.4744765!  8-42963* 

:  666  44355(5   •29540,3296   25'806975f 

i  8-732892 

BOE 

36000C 

216000000 

24-4943974!  8-434327 

:  667,  444HS9    29i57401ti53   25'826343 

!  8-737260 

60 

36120 

217081801 

24-5153013  8-439011 

;  G63   446224    298077032   25-8456H61 

!  8-741625 

60, 

362404   218167208 

24-53563^3:  8-443688 

l'  669   447501    29941.3309   25-8650341 

1  8-745985 

eo: 

36.3609   219256227 

24-5560533!  8-4483601    670  443900    300763000   25-8843582   8750340 

APPENDIX. 


23 


No. 

Square. 

Cube.    Sq.  Root.  JCubeRooulj  No. 

Square.  |   Cube. 

Sq.  Roct.  CubeRoofc 

671 

450241 

302111711 

25-9036677'  8-754691  738 

544614  401947272  27-1661554'  9-036886 

672 

451584 

303464448 

25-92296281  8-759033  739 

5461211  403533419  27-1845544  9-040965 

673 

452929 

304321217 

25-9422435  8-76333l!!  740 

547600  40522400J  27-2029410:  9-045042 

67: 

454276 

306182024 

25-9615100 

8-767719  741 

549081  406369021  27-2213152  9-049114 

07;, 

455625 

307546375 

25-9807621 

8-772053  ';  742 

550564  408518488  27-23.)f,7,-,'t  9-053183 

676 
677 

456970 
453329 

308915776 
310283733 

26-0000000  8-7763a3  ,  743  552049!  410172407  27-2530263  9-057243 
26-0192237  8-780703!  744!  553536;  411830784  27-2763(i3i  9-061310 

618 

459684 

311665752 

26-0384331!  8-7850301 

745  555025  413193625  27-2946881'  9-0653B3 

679 

461041 

313046839 

26.0576281 

8-789347  746 

556516 

415160936  27-31300C6  'J'069422 

630 

462400 

314432000 

26-0763096 

8-793659'  747 

558009 

4163327*3  27  3313007  9-073473 

631 

463761 

315321241 

26-0959767  8-7979681  ,748 

5.V9504 

413508992  27-3495337  9077  5*0 

6,5-2 
683 

46.}  124 

-166489 

31721-1568 
318611987 

26-1151297!  8-802272  749 
26-1342687  8-806572H  750 

561001 
5625JO 

420189749  27-3078644  9-OM563 
421875000  *7-3o61279  9-085603 

684 

467856 

320013504 

26-1533937 

8-810868])  751 

564o01 

423564751  27-4043792 

9-08i)63j 

685 

469225 

321419125 

26-1725047 

8815160 

752!  505504 

425*59008  27-4226184 

9-093672 

686 

470596 

322828856 

26-1916017 

8-819447 

753 

567009 

426957777!  27-4408455  9-097701 

687 

471969 

324242703 

262106848 

8-823731 

754 

568516 

423661064  27-459U6J4  9-1017«, 

688 

473344 

3*566i;672 

26-2297541 

8-828010 

755 

570025 

430363875  27-47:2633  9-10574o 

639 

474721 

327082769 

26-2483095 

8-832285 

756 

571536 

432081216 

27-4J5t542  9-l()97o7 

090 

476100 

32S509UOO 

26-2678511 

8-836556 

757 

573049 

433790093 

5J7-513633J 

«f'll3/b8l 

691 

477481 

329939371 

26-2868789 

8-840823 

753 

574564 

435519512 

U'1177t3 

692 

478864 

331373888 

26-3053929 

8-845085 

759 

576081 

437245479 

*r549jJ40 

9-1*1801 

693 

4802  19 

332312557 

26-3248932 

8-84a344 

760 

577600 

433976000 

27-5680976  9-12530^ 

i'J4 

48163C 

334255334 

26-34  387:-7 

8-853598 

761 

579121 

440711081 

695 

483025 

335702^75 

26-3628527 

8-857849 

762 

530644 

442450728 

27-6043475i  9-13330^ 

696 

484416 

337153536 

26-3318119 

8-862095 

763 

532169 

444194947 

276224546)  9-  1377^7 

697 

435809 

338608373 

26-4007576 

8.866337 

764 

533096 

445943744 

27-64o549y 

9-441787 

698 

487204 

340063392 

26-4196896 

8-870576 

765 

535225 

447697125 

27-658o334 

9-14577-1 

699 

438601 

341532099 

26-4386081 

8-874810 

766 

586756 

449455096 

276767o5o 

9-14975,5 

700 

490000 

343000000 

26-4575131 

8-879040 

767 

583289 

451217663 

27'694764a 

9-153737 

701 

491401 

344472101 

26-4764046 

8-883266 

768 

589824 

452984832 

27-7128129 

9-15<7l4 

702 

492304 

345948408 

26-4952826 

8-887483 

769 

591361 

454756609 

27-730849* 

9-16  Ibo, 

494209 

347428927 

26-5141472 

8-891706 

770 

592900 

456533000 

27-7488739 

9-165650 

704 

495616 

348913664 

265329983 

8-895920 

771 

594441 

458314011 

27-7668868 

9-16902. 

705 

497025 

35040.625 

26-5518361 

8-900130 

772 

595984 

400099648 

y-17353j 

706 

493436 

351895316 

26  5706605J  8-904337 

773 

597529 

461889917 

27-8023775 

9-  J  77544 

707 

499849 

353393243 

26-5894716  8-908539 

774 

599076 

463684824 

27-8208555 

9-18150C 

708 

501264 

354894912 

26-6082694  8-912737 

775 

600625 

465484375 

27-0330218 

9-4o545o 

709 

502631 

356400829 

26  6270539!  8-916931 

776 

602176 

467288576 

27'856776b 

y-  lO'j-iO^. 

71o 

5U4100 

35791X000 

•/6-6458u52  8-921121 

777 

603729 

469097433 

27-8747197 

9-  193347 

711 

505521 

35942543! 

26-6645833  8-925308 

778 

605284 

470910952 

27-8926514 

y-iy7*yo 

712 

506944 

3('>0944  120 

26-6833231  8-929490 

779 

606841 

47S729139 

27-9105715 

y-20i22y 

713 

508369 

362467097 

26-7020598  8-933669 

780 

474552000 

27-9284801 

9-20540-i 

714 

509796  363994344 

26-7207784  8-937843 

781 

60^961 

476379541 

27-94o3772  9-XO9O90 

715 

511225  365525375 

$••7394839]  8-942014 

782 

611504 

470211768 

27-yo42029i  y-zl30*5 

716 

512056  367061696 

26-7581763!  8-946181 

783 

613089 

48C048687 

27'«'821b7i 

y-2l095v 

717 

5140o9  368601813 

26-77635571  8-950344 

734 

614656 

481890304  28-0000000 

y-*2o8/o 

718 

515521  370146232 

26-7955220  8-954503 

785 

616225 

4837366251  280l7o515 

y-*2479i 

719 

516961'  371694959 

26-8141754  8-958658 

786 

617796 

485587656  28'0356-jl5 

y-**37o7 

720 

518400  373248000 

26-8328157  8-962809 

787 

619369 

487443403 

V8-0535203 

9-*^2ol9 

721 

519841  374805361!  26'8514432'  8'966957 

780 

620944 

489303372 

28-0713377 

y-*u>o52o 

5212341  37636704*1  26'8700577!  8-971101 

789 

632521 

491169069 

28-oayi433 

9-*io-iito 

7*J» 

522729  377933067 

26-8886593!  8-975241 

790 

6*4100 

493039000 

23-1069380 

9**44«>3t; 

72-i 

524176  3795L3424 

26-90724811  8-979377 

791 

625681 

494913671 

28-1247-2^ 

y-*io*i>4 

725 

5*50*5  331078125 

26-9258240!  8-933509 

792 

627264 

496793088 

28-1424940 

y-2o21«H> 

721 

527076'  382657176 

26-9443872  8-9a76:-.7  791- 

628849 

493677*57 

23-1602557 

9"*5oo** 

727  528529  3342405s3 
723  529984  385828352 

26-9629375  8-991762  79; 
26-9814751;  8-995883  795 

630436 

632025 

500566184 
502459875 

28-1780050  9-*ay9ii 

23-  1  95744-4J  y-*6.>797 

729 

531441  337420489 

27-OOOOOCO  9-000000  796 

633610 

504353336 

23-2l3472u  y-aO/08« 

73<J 

532900  389017UOO 

27-0185122  9-004113  79,' 

635209 

506261573 

28-234l»o4  9--471:wi» 

731 

534361  390617891 

27-0370117  9-0032*3  79s 

636804 

508169592 

28-2488930  y-zV»4i>a 

535o2-i  392223160 

27-0554985  9-012329  799 

638401 

510082399 

28-*66588l  y-i7S)3wo 

7B5 

27-0739727  9-016431  MOO 

640000  512000000 

28-284271*1  9-2oil7o 

734 

53^7:-6  3->5-l4<>yi>4  27-0924344  9-020529 

801 

641601!  513922401 

23-3019434J  y-to/W^ 

735 

54U*25  397065375  27.1108834  9-b2;624  30* 

643-lOi1  515349608 

*3'3iy^o45  y'*yO"Oi 

5-llt.y..  39^83^56!  27-1293199  9-0^715!  603 

644809  517781627 

28-o37*5iO  y-*9470. 

7'-.7 

5431t>9  400315553'  27-147743.'  9V32302|;  804 

646416]  519718464 

*8-354o930)  y-298o2t 

APPENDIX. 


No. 

Square.)   Cube.     Sq.  Root.  CubeRooU  No.  Square. 

Cube.   1  Sq.  Rent. 

JubeRoot. 

805 

648025  521660125  283725219 

9-302477 

872  760384 

663054848 

29-5296461 

9-553712 

806 

649636 

5236uG616  283901391 

9-306323 

873  762129 

665338617 

29-5465734 

9-557363 

807 

651249  525557943;  28-4077454 

9-310175 

874  763376 

667627624 

29-5634910 

9-561011 

808 

652864  527514112;  284253408 

9-314019! 

875 

765685 

669921875 

29-5803989 

9-564656 

809 

654181  529475129  28-4429253 

9-317860 

876  767376 

072221376 

29-5972972 

9-568298 

SlOl  656100!  53144lOOOi  28-4604981) 

9-321697J 

877  769129 

674526133 

29-6141858 

9-571938 

811 

657721  533411731i  28-4780617 

9-325532  878|  770884 

676836152 

29-6310648 

9-575574 

812 

65U344  535387328!  28-4956137 

9-329363 

879  772641 

679151439 

29-6479342 

9-579208 

813 

660969  537367797i  23-5131549 

9-333192] 

880  774400 

681472000 

29-6647939!  9-532340 

814 

662596  539353144  28-5306852 

9-337017 

&81  776161 

683797841 

29-6816442  9-;  86468 

815 

664225 

541343375  23-5482048 

9-340839 

832 

777924 

686128968  2:!.6984843  9-590094 

816 

605356 

543338496;  23-5657137 

9-344657! 

SS3 

779639 

688465387J  29-7153159!  9-593717 

817 
818 

607489 
669124 

545333513;  28-5832119 
547313432!  '28-6006993 

9-3i8173i 
9-352286 

8>i 
tfe» 

781456 
783225 

69U807101  29-7321375  9-597337 
693154125  29-7439496'  9-600955 

819 

670761 

549353259)  28-6181760 

9-356095 

686 

78499" 

695505456  29-7657521  9-60457u 

820 

672400 

551363000;  28-6356421 

9-359902 

887 

780769 

697864103|  29-7825452 

9-608182 

821 

674041 

553337661;  28-6530976 

9363705; 

888 

788544 

7002270721  29-7993239  9-611791 

822 

675684 

555412248  28-6705424 

9367505 

889  790321 

70-^595369 

29-8161030,  9-615391- 

8i3 

677329 

557441767  23-6379766 

9-371302 

890 

7b2100 

704969000 

29-8323678  9-619(X)2 

824 

673976 

559476224  23-7054002 

9-375096 

8'.)1 

793881 

707347971 

29-8496231!  9-0220  U 

825 

680625 

561515625;  28-7228132 

9378887J 

892 

795664 

709732288 

29-8663690J  9-62620^ 

826 

632276 

503559976  28-7402157 

9-332675 

3.13 

797449 

712121957 

29-8831056I  tJ-C2J7'J7 

827 

633929 

565609283  28-7576077 

9-336460! 

894 

799236 

714516934 

29-899838S  9-63T91 

823  685581 

567663552  28-7749891 

9-390242 

895 

801025 

716917375 

2J-J1655U6  9-636  JS) 

829  687241 

569722789|  28-7923601 

9-394021 

896 

802816 

719323136 

2i>>332591  964056. 

830  6889  '0 

571787000  28-809720f 

9-3.7796 

897 

804609 

721734273 

2J-9499533  9-04415! 

831  690561 

5738661911  28-82707C6I  9-401569 

898 

806404 

724150792 

2'J-9666481  9-647737 

832  69-224 
833  693889 
834  6-J5556 
835  697225 

575930368  28-8-144102  9-405339! 
578009537  28-8617394  9-409105 
5S0093704  28-8790582  9-412869 
582182875  28-8963606!  9-416630 

899 

900 
901 
'JO-J 

808201 
810UOO 
811801 
813604 

72657269V 
729000000 
731432701 
733870808 

29-9.«>33*>7i  S-051317 
30-OOOOOOOj  9-054334 
30-0106620!  9-6.J3168 
30-0333148;  'J-t:»i:!;-l( 

o36;  693896 

584277056  28-91360461  9-420387 

903 

815409 

736314327 

30-0499534!  9-665610 

837  700569 

586376253  28-9309523J  9-424142 

904 

817216 

738703264 

30-0065  i)23  9-GO-J170 

838  702244 

5^8480472  28-9482297  9-4278^4 

^o:> 

819025 

741217625 

30-0832179]  9-67274U 

839  1  7^3921 

590539719  23-9651967  9-431642, 

906 

820836 

74:J6774K> 

30-099833  j:  9-6763J2 

8401  705600 
841  707^81 

5J2704000  28-98275351  9-435388 
594823321i  29-0000000  'J'439131 

9ffi 
908 

822641 
824464 

74614264-3 
748613312 

30-1164407!  lj-67:86n 
30-13303831  9-633417 

842'  708;  64 

596947688  29-01723631  9-442370 

909 

826281 

751089429 

30-1496269:  \)-(,^<M« 

813:  710649  599077107!  29-03446231  9-446607 
844  71i336i  601211584!  29-0510781|  9'450341 

911 

911 

828101 
829921 

753571000 
750053031 

30-  1662063]  9-690521 

30-1827765i  9-69406i 

845  714125!  603351  125  i  29-0688837  9-454072 

912  83174- 

758550528 

30-1993377  9-697615 

846  715716  605495736!  29-0860791 

9-457800 

913  83356t 

761048497 

30-2158399  9-701158 

847i  717409!  607645123  29-1032G44 

9-461525 

911  835396 

762551944 

30-2324329;  9-70169fc 

843!  719104|  609.300192  2J-12J43'Jf 

9-465247 

915J  837  22f 

766C60S75 

30-2489669  9-708i37 

849  7208011  611960049!  29-13700H 

9-463966 

yie 

839056  768575296 

30-2654919 

9-711772 

850!  7:22500  6141250001  29-1547595 

9-47^682 

91  ; 

840889!  771095213 

30-282007i) 

9-715305 

85  ll  724201 

6162.50511  29-1719042 

9-4763J6 

913 

8-42724  773620632 

302J85148 

9-718835 

852i  725904 

618470203  29  18903  JO  9'480K.6 

yi9 

844561!  770151559 

30-3150128 

9-722303 

853  72760. 

620050477  29-2061637  9-483814 

92i 

846400!  773638000 

30-3315018 

9-725388 

854 

7293  If 

62:2835364  29-2232784  9-487518 

bit] 

848241  781229%! 

30-3479818 

9-729411 

855  7310:5 

6250*6375  29-2403830:  9'491220 

922 

8500841  783777448 

30-3644529 

9-732931 

856  73273f 

627222016  29-2574777.  9-494919 

923  8519291  78C330467 

303809151 

9-736448 

857!  73444H 

029422793  29-2745623  9-498615 

924 

853776!  788839024 

30-3J73683 

9-739963 

858i  736161  63102.5712  2929)6370  9'5023.i8 

925 

855625  791453125 

30-4138127 

9-743176 

859!  737881  633339779,  29-3087018  9'5059ii8  926 

857476  79402277f 

3J-4302481 

9-746986 

860  73J0001  0;;6J56000  2J  3-57563  9-509685  92; 

8593291  796597983 

30-4466747 

9-750493 

861;  741321  G33,;7733l!  29-3425015  9-513370 

9J- 

861184 

79917375;; 

30-4630924 

9-753998 

8621  743(144  6luK,>3923;  z9  351>83G5  9-517051 

92; 

^63041 

801765039 

30-479501S 

9-7575CX. 

803  7-4471)9!  042735047;  293708616  9-520730 

93i 

864900:  8C4357<KX)  30-49590141  9-761000 

864  ;  746496J  614972514;  i;9  393376J  9-524406 

'.-:;l 

866761;  Su6l;o-5491  30-5122926!  9-764497 

8ti5  743.25  64:2146^5  *y-4103323  9'5280;9 

9SI 

86*?24.  8095575G8  30-52867501  9-707992 

»6f>:  741-956  649461895  2v427S779  9-531750 

13: 

87J4-r.»j  812j'J6237|  30-5450437|  9-771484 

867!  75168.,  651714363  29-4448637  9-535417 

934 

872356-  8147305041  30-5614136;  9-774974 

868|  753r24 

653J72u32  29-4018397  »'5b9082 

Mb  874225  817400375!  3'>57776-»7i  9-778462 

869  755101  656234909  2J-4788059  9'542744  9o6j  87601)6  820U25y56|  30-5.;4ini  9781947 

»70  756901 

6o35o3000  29-4V57624  9  546403  i  9o'7  877909  822056'.  531  3NJki455:;  9-7354i9 

8711  75804 

600776311:  2j-51a7o-.'l  9-;'5J059,i  9Jo  879844  82529367«|  3J-0267857>  9-73-9O9; 

APPENDIX. 


25 


Mo. 

Square.    Cube. 

Sq.  Root. 

CubeRoot. 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

939|  881721 

827936019 

30-6431069 

9-792386 

97( 

940900 

912673000 

31-1448230 

9-898983 

940  883600 

830584000 

30-6594194 

9-795361 

971 

942841 

915498611 

31-1608729 

9-9023-J3 

941  885481 

833237621 

30-6757233 

9-799334 

972 

944784 

918330048 

31-1769145 

9-905782 

942  887364 

835896888 

30-6920185 

9-802804 

973 

946729)  921167317 

31-1929479 

9-909173 

943  889249 

838561807 

30-7083051 

9-806271 

974 

948676 

924010424 

31-2089731 

9-912571 

944|  891136 

841232334 

30-7245830 

9-809736 

975 

950625 

926859375 

31-2249900 

9-915962 

(J45i  893025 

843308625 

30-7408523 

9-813199 

976 

952576 

929714176!  31-2409987 

9-919351 

946!  894916 

846590536 

30-7571!  30 

9-816659 

977 

954529 

932574833  31-2569992 

9-92273S 

94?  896809 

849278123 

30-7733651 

9-820117 

978 

956484 

935441352 

31-2729915 

9-926122 

948  898704 

851971392 

30-7896086 

9-823572 

979 

958441 

938313739 

31-2889757 

9-929504 

949  900601 

854670349 

30-8058436 

9-827025 

980 

960400 

941192000 

31-3049517 

9-932884 

950  902500 

857375000 

30-8220700 

9-830476 

(.),81 

962361 

944076141 

31-3209195 

9-936261 

951|  904401 

860085351 

30-8382879 

9-833924 

Si82 

964324 

946966168 

31-3368792 

9-939636 

952 

906304 

862801408 

30-8544972 

9-837369 

9s;i 

966289 

949862087 

31-3528308 

9-943009 

953 

908209 

865523177 

30-8706981 

9-840813 

984 

968256!  952763904 

31-3687743 

9-916380 

954 

910116 

868250664 

30-8868904  9-844254 

985 

970225  955671625 

31-3847097 

9-949748 

955 

912025 

870983875 

30-9030743 

9-8476:12 

981 

972196  958535256 

31-4006369 

9-953114 

95G 

913936 

873722816 

30-91  924  97  j  9-8511^8 

1)87 

974169  961504803 

31-41(15561|  9-956477 

957 

915849 

876467493 

30-9354166  9-854562 

2d8 

976144  9644302"2 

31-4324673  9-959839 

958 

917764 

879217912 

30-9515751 

9-857993 

989 

97812  1!  967361669 

31-4483704  9-963  19S 

959 

919681 

881974079 

30-9f>7725l|  9-861422 

'.)'.)< 

980100 

970299000 

31-4642654  9-966555 

l.'60 

921SOO 

884736000 

30-9^386681  9-864848 

'J'Jl 

982081 

973242271 

31-4801525  9-969909 

961 

923521 

887503681 

ai-0000000  9-868272 

'J92 

984064 

976191488 

31-4960315!  9-973262 

962 

925444 

890277128 

31-0161248 

9-871694 

993 

986049 

979146657 

31-5119025!  9-976612 

963 

927369 

893056347 

31-0322413  9-875113 

994 

988036 

982107784 

31-5277655  9-979960 

9G4 

929296 

81)5841344 

31-0483494 

9-878530 

995 

990025 

985071875 

31-5436206!  9-933305 

965 

931225 

898632125 

31-0644491 

9-881945 

99C 

992016 

988047936 

31-55946771  9-9866  la 

9661  933156 

901428696 

31-0805405 

9-885357 

997 

994009 

991026973 

31-5753068  9-989990 

967!  935089 

904231063 

31-0966236 

9-883767 

998 

996004 

994011992 

31-5911380  9-99332* 

963 

9G9 

937024  907039232 
93896  l|  909853209 

31-1126984 
31-1287648 

9-892175 
9-895580 

999 

loot 

998001!  997002999  31-6069613-  9-996666, 
1000000  '1000000000:  31-6227766  10'OOOOoOj 

The  following  rules  are  for  finding  the  squares,  cubes  and  roots,  of 
numbers  exceeding  1,000. 

To  find  the  square  of  any  number  divisible  without  a  remainder. 
Rule. — Divide  the  given  number  by  such  a  number,  from  the  forego- 
ing table,  as  will  divide  it  without  a  remainder  ;  then  the  square  of  the 
quotient,  multiplied  by  the  square  of  the  number  found  in  the  table, 
will  give  the  answer. 

Example  -  What  is  the  square  of  2,000  ?     2,000,  divided  by  1 ,000, 
a  number  lound  in  the  table,  gives  a  quotient  of  2,  the  square  of  which 
is  4,  and  the  square  of  1,000  is  1,000,000,  therefore  : 
4  X  1,000,000  =-  4,000,000  :  the  Ans. 

Another  example. — What  is  the  square  of  1,230  ?  1.230,  being  dl 
vided  by  123,  the  quotient  will  be  10,  the  square  of  which  is  100,  and 
the  square  of  123  is  15,129,  therefore  : 

100  X  15,129  —  1,512.900 :  the  Ans. 

To  find  the  square  of  any  number  not  divisible  without  a  remainder. 
Rule. — Add  together  the  squares  of  such  two  adjoining  numbers,  from 
the  table,  as  shall  together  equal  the  given  number,  and  multiply  the 
sum  by  2  ;  then  this  product,  less  1,  will  be  the  answer. 

Example. — What  is  the  square  of  1,487  ?  The  adjoining  numbers, 
743  and  744,  added^together,  equal  the  given  number,  1,487,  and  the 
square  of  743  =  552,049,  the  square  of  744  =  553,536,  and  these 
added,  —  1,105,585,  therefore  : 

1,105,585  X  2  =  2,211,170 —  l  —  '2,211,169:  the  Ans. 

To  find  the  cube  of  any  number  Divisible  without  a  remainder. 
Rule. — Divide  the  given  number  by  sucn  a  number,  from  the  forego 


26  APPENDIX. 

ing  table,  as  will  divide  it  without  a  remainder  ;  then,  the  cube  of  tne 
quotient,  multiplied  by  the  cube  of  the  number  found  in  the  table,  will 
give  the  answer. 

Example. — What  is  the  cube  of  2,700  ?  2,700,  being  divided  by  900, 
the  quotient  is  3,  the  cube  of  which  is  27,  and  the  cube 'of  900  is 
729,000,000,  therefore  : 

27  X  729,000,000  —  19,683,000,000 :  the  Ans. 

To  find  the  square  or  cube  root  of  numbers  higher  than  is  found  in  the 
table.  Rule. — Select,  in  the  column  of  squares  or  cubes,  as  the  case 
may  require,  that  number  which  is  nearest  the  given  number  ;  then 
the  answer,  when  decimals  are  not  of  importance,  will  be  found  di- 
rectly opposite  in  the  column  of  numbers. 

Example. — What  is  the  square-root  of  87,620  ?  In  the  column  of 
squares,  87,616  is  nearest  to  the  given  number  ;  therefore,  296,  im- 
mediately opposite  in  the  column  of  numbers,  is  the  answer,  nearly. 

Another  example. — What  is  the  cube-root  of  110,591  ?  In  the  co- 
lumn of  cubes,  110,592  is  found  to  be  nearest  to  the  given  number  ; 
therefore,  48,  the  number  opposite,  is  the  answer,  nearly. 

To  find  the  cube-root  more  accurately.  Rule. — Select,  from  the  co- 
lumn of  cubes,  that  number  which  is  nearest  the  given  number,  and 
add  twice  the  number  so  selected  to  the  given  number  ;  also,  add  twice 
the  given  number  to  the  number  selected  from  the  table.  Then,  as 
the  former  product  is  to  the  latter,  so  is  the  root  of  the  numbar  selected 
to  the  root  of  the  number  given. 

Example. — What  is  the  cube-root  of  9,200  ?     The  nearest  number 
in  the  column  of  cubes  is  9,261,  the  root  of  which  is  21,  therefore  : 
9261  9200 

2  2 


18522      18400 
9200       9261 


As  27,722.  is  to  27,661,  so  is  21  to  20-953  4-  the  Ans. 

Thus,  27661  x  21  =  580881,  and  this  divided  ly  27722  =  20-953  -f 

To  find  the  square  or  cube  root  of  a  whole  number  with  decimals. 
Rule. — Subtract  the  root  of  the  whole  number  from  the  root  of  the  next 
higher  number,  and  multiply  the  remainder  by  the  given  decimal ;  then 
the  product,  added  to  the  root  of  the  given  whole  number,  will  give  the 
answer  correctly  to  three  places  of  decimals  in  the  square  root,  and  to 
seven  in  the  cube  root. 

Example. — What  is  the  square-root  of  11*14?  The  square-root  of 
11  is  3*3166,  and  the  square-root  of  the  next  higher  number,  12,  is 
3*4641 ;  the  former  from  the  latter,  the  remainder  is  0-1475,  and  this  by 
0-14  equals  0*02065.  This  added  to  3*3166,  the  sum,  3*33725,  is  the 
square  root  of  11*14. 

To  find  the  roots  of  decimals  by  the  use  of  the  table.  P.ule. — Seek  for 
the  given  decimal  in  the  column  of  numbers,  and  opposite  in  the  col- 
umns of  roots  will  be  found  the  answer,  correct  as  to  the  figures,  but  re- 
quiring the  decimal  point  to  be  shifted.  The  transposition  of  the  deci- 
mal point  is  to  be  performed  thus  :  For  every  place  the  decimal  point  is 
removed  in  the  root,  remove  it  in  the  number  two  places  for  the  squart 
root  and  three  places  for  the  cube  root. 


APPENDIX.  27 

Examples. — By  the  table  the  square  root  of  86-0  is  9-2736,  conse- 
quently, by  the  rule  the  sqjare  root  of  0-86  is  0-92736.  The  square 
root  of  9'  is  3-,  hence  the  square  root  of  0"09  is  0'3.  For  the  square 
root  of  0-0657  we  have  .0-25632;  found  opposite  No.  657.  So,  also, 
the  square  root  of  0-000927  is  0-030446,  found  opposite  No.  927.  And 
the  square  root  of  8*73  (whole  number  with  decimals)  is  2-9546,  found 
opposite  No.  873.  The  cube  root  of  0-8  is  0-928,  found  at  No.  800  ; 
the  cube  root  of  0-08  is  0-4308,  found  opposite  No.  80,  and  the  cube 
root  of  0-008  is  0-2,  as  2-0  is  the  cube  root  of  8*0.  So  also  the  cube 
root  of  0-047  is  0-36088,  found  opposite  No.  47. 


RULES  FOR  THE  REDUCTION  OF  DECIMALS. 

To  reduce  a  fraction  to  its  equivalent  decimal.     Rule. — Divide  the 
numerator  by  the  denominator,  annexing  cyphers  as  required. 
Example. — What  is  the  decimal  of  a  foot  equivalent  to  3  inches  ? 
3  inches  is  fo  of  a  foot,  therefore : 
^  ...  12)  3-00 

•25  Ans. 

Another  example. — What  is  the  equivalent  decimal  of  £  of  an  inch! 
J  .  .  .  8)  7-000 


•875  Ans. 

To  reduce  a  compound  fraction  to  its  equivalent  decimal.  Rule.  —  In 
accordance  with  the  preceding  rule,  reduce  each  fraction,  commencing 
at  the  lowest,  to  the  decimal  of  the  next  higher  denomination,  to  which 
add  the  numerator  of  the  next  higher  fraction,  and  reduce  the  sum  to 
the  decimal  of  the  next  higher  denomination,  and  so  proceed  to  the  last  ; 
and  the  final  product  will  be  the  answer. 

Example.  —  What  is  the  decimal  of  a  foot  equivalent  to  5  inches,  f 
and  -j-L-  of  an  inch. 

The  fractions  in  this  case  are,  1  of  an  eighth,  £  of  an  inch,  aud  *5  of 
a  foot,  therefore  : 


•5 

-  eighths. 


8)  3-5000 
•4375 


inches. 


Al'i'liNUlX. 


The  process  may  be  condensed,  thus  ;  write  the  numerators  of  the 
given  fractions,  from  the  least  to  the  greatest,  under  each  other,  and 
place  each  denominator  to  the  left  of  its  numerator,  thus  : 


3-5000 
5-437500 

•453125  Ans. 

To  reduce  a  decimal  to  its  equivalent  in  terms  of  lower  denominations. 
Rule. — Multiply  the  given  decimal  by  the  number  of  parts  in  the  next 
less  denomination,  and  point  off  from  the  product  as  many  figures  to 
the  right  hand,  as  there  are  in  the  given  decimal ;  then  multiply  the 
figures  pointed  off,  by  the  number  of  parts  in  the  next  lower  denomina- 
tion, and  point  off  as  before,  and  so  proceed  to  the  end ;  then  the  seve- 
ral figures  pointed  off  to  the  left  will  be  the  answer. 

Example. — What  is  the  expression  in  inches  of  0-390625  feet? 
Feet  0-390625 

12  inches  in  a  foot. 


Inches  4-687500 

8      eighths  in  an  inch. 

Eighths  5-5000 

2  sixteenths  in  an  eighth. 

Sixteenth  1-0 

Ans.,  4  inches,  |j-  and  -J-g. 

Another  example. — What  is  the  expression,  in  fractions  of  an  inch, 
of  0'6875  inches? 

Inches  0-6875 

8  eighths  in  an  inch.  . 

Eighths  5-5000 

2        sixteenths  'in  an  eighth. 

'  Sixteenth  1-0 

Ans.,  £  and  ^ 


TABLE  OF  CIRCLES. 

(From  Gregory's  -Mathematics.) 

From  t'.iis  table  may  be  found  by  inspection  the  area  or  cireumie" 
.rence  of  a  ciicie  of  any  diameter,  and  the  side  of  a  square  equal  to  the 
area  of  any  given  circle  from  1  to  100  inches,  feet,  yards,  miles,  &c. 
If  the  given  diameter  is  in  inches,  the  area,  circumference,  &c.,  set 
opposite,  will  be  inches  ;  if  in  feet,  then  feet,  &c. 


Side  oi 

>.  ;••  .•: 

Biaai. 

Area. 

1      Circum. 

equal  sq. 

Diam. 

Area. 

|     Circuen.     |   equal  »,. 

•tt 

•04908 

•78539 

•22155 

•75 

90-76257 

3377212 

9526-.I3 

•5 

•19635 

1-57079 

•44311 

11- 

95-033  1  7 

1       34-55751 

9-71-il.' 

*75 

44178 

!        2-33619 

•66467 

•25 

99-40195 

1      35-34291 

9-971X15 

j. 

•7353J 

3-14159 

•88622 

•5 

103-86890 

Sir  12331 

10-19160 

•25 

1-22718 

!        3-92699 

1-10778 

*7S 

108-43403 

36-91371 

10-41316 

-5 

1-76714 

4-71238       1-32934 

12- 

11309733 

37-69911 

10-63472 

•7.-) 

2-4052? 

5-497731       1-5508? 

•25 

117-85381 

i      33-43451 

10-85S27 

2- 

3-1415' 

6-23318 

1-77245 

•5 

122-7184G 

i      3926990 

11-07783 

•25 

3-'.!7t;o7 

7-06858 

1-99401 

•75 

127-67628 

40  05530 

11  -2993  J 

*5 

4-90879 

7-85393 

2-21556 

13- 

132-73228 

40-84070 

11-52095 

•75 

5-93957 

8-63937 

2-43712 

•25 

137-83646 

41-62610 

11  -7425  > 

3 

7-06853 

942477 

2-65363 

•5 

143-13881 

42  -4  11;  50 

ii-9(um» 

*25 

82,576 

10-21017 

2-88023 

•75 

148-48994 

45-19689 

12-KsSCJ 

•5 

9-62112 

10-99557 

3-10179 

14- 

153-93804 

4348299 

1240717 

•73 
4- 

ll-0-14f,6i       11-78097       332335 
12-566371       12-56637i      3-5(490 

•25 
•5 

159-43491!       44-76769 
165-  12996!      45-55309 

12-85029 

•25 

14-18625 

13-35176|       3-76646 

•75 

170-87318 

46-333491 

1307184 

•5 

15-90431 

14-13716       3-98802 

15- 

176-71458 

47-  123  -(3 

13-2.'34J 

17-72054 

14-92256        4-20957 

•25 

182-65418 

47-90923 

13-51196 

5-'J         19-63495 

15-70796!      4-43113 

-.r) 

188-69190 

48-694R8J 

1373.151 

•25         21-64753 

16-49336!       4-05269 

•75 

191-82783 

49-4ax>3i 

13-95307 

•5 

23-75829 

17-27875        4-87424 

16- 

201-06192 

50--265I8 

14-17963 

•75 

2596722 

18-06415       5-09500 

•25 

207-39420 

51-05088 

14-  10  11s 

6- 

28  -27433 

18-84J551      5-31736!       -5 

213-82464 

51-33-J27 

1462274 

•25 

30-67961 

19-63495        5-53891J        -75 

220-35327 

5262167! 

14v»4»30 

•5 

38-18307 

•2I-12H33        576il47     17- 

226-98006 

53-40707 

13-0(,5\5 

•75 

35-78470 

21-20575        5-932031        -25 

233-70504 

54-  19247 

15-28741 

7- 

33-48456 

21-99111 

620353        -5 

24*52818 

54-97737 

1550897 

25 

41-28249 

27-77654 

6-42514        -75 

847-44850 

55  76326J 

15-73053 

•£ 

44-17864 

23-56194        6-616701:  18- 

264-46900 

55-54866 

15-9520,s 

•75 

47-17297 

•2134734       6-86825}'      -25 

266-58667 

57-33406 

1C,-17:{64 

8- 

50-26548 

25-13274       7-08981!       -5 

26880252 

53-11946 

1639519 

25 
•5 

•75 
9- 
•25 
•5 

5345616 

51V74501 
60-13201 
6361725 
6720063 

70-SS-213 

2V91313       7-31137, 
•26-70353        753292] 
27-48893        7'75  4  H 
28-27433        7-97604 
29-0597*       8-197591 
29-84513:       8-41915 

•75 
19- 
•25 
•5 

•75 
20- 

276-11654 
233-52873 
291-03910 
298-64765 
306-35437 
314-15926 

53-904  3<i 

8047569 

r,i-2f.li;5 
(•,•2-04645 
62-8318j| 

1(5-6  1075 
16-83331 

17-2814J 

17-5>2;)8 
17-72153 

•75         74-66191 

30-63052 

8-64071 

322-06233 

63-61725; 

1  7-941109 

10-              7853931        31-41592       8-86226  i       -5          330-06357 
•25\        82-515391      32-2013-2       9-083S2|!      -75       33S-  16-299 

M  -40264  ! 
65-183OJ 

13-10765 
18-:W920 

•5  1         86-59014 

32-986721       9-30533:    21-           346-36053 

6597314 

18-01076 

30 


APPENDIX. 


Diam. 

Area. 

|      Side  of 
Circum.          equal  sq. 

Diam. 

Area. 

Circum. 

Si.le  of 
eoual  sq. 

21-25 

354-65635 

66-75884      18-83232 

38- 

1134-114941     119-38052 

3367662 

•5 

363-05030 

67-54424 

19-05387 

•25 

1149-08660      120-16591 

33-89817 

•75 

371-54241 

68-32964      19-27543 

•5 

1164-15642      120-95131 

34-11973 

22- 

380-13271 

69-11503'     19-49699 

•75 

1179-32442      121-73671 

34-34129 

•25 

388-82117 

69-90043      19-71854 

39- 

1194-59060      122-52211 

34-56285 

•5 

397-60782 

70-68583      19-94010 

•25 

1209-95495'     123-30751 

34-78440 

•75 

406-49263 

71-47123!     20-16166 

•5 

1225-41748'     124-09290 

35-00596 

23- 

415-475K2 

72-25663     20-38321 

•75 

12-10-97818      124-878301     35-22752 

•25 

424-55679 

73-04202     20-60477 

40- 

1256-637041     125-663701     35-44907 

•5 

433-73613 

73-82742'     20-82633 

•25 

1272-39411      126-449  10!     35-67063 

•75 

443-01365 

74-61282     21-04788 

•5 

1288-24933 

127-23450     35-89219 

24- 

452-38934 

75-39822     21-26944 

•75 

1304-20273 

128-019901     36-11374 

•25 

461-8S1320 

76-18362     21-49100 

41- 

1320-25431 

123-805291     3633530 

•5 

471-43524 

76-96902     2171255 

•25 

1336-40406 

129-590691    36-55686 

•75 

481-10546 

77-75441     21-93411 

•5 

1352-65198 

130-37609!     36-77841 

25- 

490-87385 

78-53981      22-15567 

•75 

1368-99808 

131-16149i     36-99997 

•25 

500-74041 

79-32521  i     22-37722 

42- 

1385-44236 

131-94689     37-22153 

•5 

510-70515 

80-11061J     22-59878 

•25 

1401-98480 

132-73228!     37-44308 

•75 

520-76806 

80-89601     22-82034 

•5 

1418-62543 

133-51768!     3766464 

26- 

530-92915 

81-68140!     23-04190 

•75 

1435-36423 

131-30308!     3788620 

•25 

541-18842 

82-466801     23-26345 

43- 

1452-20120 

135-08348!     33-10775 

•5 
•75 

551-54586 
562-00147 

83-25220:     23-48501 
84-03760     23-70657 

•25 

1469-13635 
1486-16967 

13587383!     3332931 
136-65928!     3855087 

27- 

572-55526 

84-82300     23-92812 

•75 

1503-30117 

137-44467!     33-77242 

•25 

583-20722 

85-60839     24-14968 

44* 

1520-53084 

138-23007;     38-99S..8 

_    -5 

593-95736 

86:39379!     24-37124 

•25 

1537-85869 

l?9-01547i     3'J-21554 

•75 

60T-80567 

87-17919:    24-5<J279 

•5 

1556-23471      139-80087;     39-43709 

28- 

615-75216 

87-964591    24-81435 

•75 

1572-80890      140-586271     39  !',:.  -.(',:, 

•25 

626-79682 

8874999!     25-03591 

45- 

159043128 

141-371661     39-88021 

•5 

637-93965 

89-53539     25-25746 

•25 

1608-15182 

142-157061     40-10176 

•75 

649-18066 

90-32078     25-47i>02 

•5 

1625-97054 

142-9  12461     40-3233-2 

29- 

66051985 

91-10618      25-70053 

•75 

164388744 

143-72786!     40-54188 

•25 

671-95721 

91-89153!     25-92213 

46- 

1661-90251 

144  51326!     40-76643 

•5 

683-49275 

92-676981     26-143o9 

•25 

168O01575 

145-29866]     40-(.W7-.ni 

•75 

695-12646 

93-46231!     26-3lo-.i.-> 

•5 

1698-22717 

146-08405J     41-2;)955 

3D- 

706-85834 

94-247771     26-58680 

•75 

1716-53677 

146-86945!     41-43110 

•25 

71868840 

95-03317J     26-80836 

47- 

1734-94454 

147-654S5     41-652-iR! 

•5 

7  30-6  1664 

95-81857J     27-02992 

•25 

1753-45048 

14844025 

4187422; 

•75 

7-!2  64305 

96-60397!     27-25147 

•5 

1772-05460 

149-22565 

42-09577 

31- 

751-76763 

97-38937:    27-47303 

•75 

1790-75639 

150-01104 

4231733 

•25 

7o6-99039' 

98-17477     27-6945'J 

48- 

1809  55736 

150-79644 

42-53889 

•5 

779-31132 

9896016;    27'91614 

•25 

1828-45601 

151-58184 

42-76044 

•75 

791-73043 

99-74556     28-13770 

•5 

1847-45232 

152-36724 

42-98200 

32-4 

804-24771 

100-53096 

28-35926 

•75 

1866-54782 

153-15264 

43-20356 

816-86317 

101-31636 

23  -59081 

49- 

1885-74099 

153-93804 

4342511 

•5 

829-57631 

102-10176 

28-J-0237 

•25 

1905-83233 

154-72343 

43-64667 

•75 

842-38861 

102-88715 

29  023^3 

•5 

1924-42184 

155-50883 

43-86823 

33- 

855-29859 

10367255 

29-21548 

•75 

1943-90954 

156-29423 

44-H8978 

•25 

868-3C675 

104-45795 

29-46704 

50- 

1963-49540 

157-07963 

4131134 

•5 

831-41308 

105-24335 

29-68860 

•25 

1983-17944 

157-965;;3 

44-53290 

•75 

894-61759 

106-02875 

29-91015 

•5 

2002-96166 

153-65012 

44-75445 

34- 

907-92027 

106-81415 

30-13171 

•75 

2022-84205 

159-43532 

44-97601 

25 

921-32113 

107-59954 

30-35327 

51- 

2042-82062 

160-22122 

45-19757 

•5 

934-82016 

108-33494 

30-57482 

•25 

2062-8973:J 

1610(662 

45-41912 

•75 

94841736 

109-17034 

30-796'W 

•5 

2083-07227 

161-79202 

45-64068 

35- 

962-11275 

109-95574 

31-01794 

•75 

2103-34536 

162-57741 

43-85224 

•25 

975-90630 

110-74114 

31-23949 

52- 

2123-71663 

163-36281 

4tri>;'3'30 

•5 

989-79803 

111-52653 

31-46105 

•25     2141-18607 

164-14821      46-3i)535 

•75 

1003-78794 

112-31193 

31-68261 

•5       2164-753C8 

184-93381!     4652691 

36- 

1017-87601 

113-09733 

31-90416 

•75 

2185-41947 

165-71901  1     46-74847 

•25 

1632-06227 

11  3-88273 

32-12572 

53-          220(5  -18344 

166-50441  j     4IV'/7.X)2 

•5 

1046-34670 

114-66313 

32-34728 

•25 

2227-04557 

167-28980     47-19158 

•75 

1060-72930 

115-45353 

32-56883 

•5 

2248-00589 

168-07520     47-41314 

37- 

1075-21008 

1  16-238i;2 

32-7903.) 

•75     2269-06433 

168-86060     47-63469 

•25 

1089-78903 

117-02432 

33-01195 

54-     |     2290-22104 

169-64600 

47-85625 

•5 

1104  -466  If 

117-80972 

33-23350 

•25  1    2311-475*8 

170-43140 

48-0778'l 

•75 

11  19-2414"      118-59572 

33-45506 

•5       2332-82889      171-216791    48-29936 

APPENDIX. 


31 


Di;.rn. 

Ana, 

1      Circuic. 

3i.le  of 

equal  sq. 

[  Diam. 

Area. 

Circum. 

1     fcJeoT 
equal  ^. 

55- 

2354-2800- 
2375-82  344 

172-00219 
i     172-73759 

43-52092 
48-74248 

71-5 
•75 

i     4015-15176 
404327833 

2-24-62337 
225-40'.!27 

6336522 
63-58678 

•25 
•5 
•75 
56- 

23..7-4769H 
2419-22-26^ 
2141-00657 
24C3-00861 

73-57299 
74-35333 
>       75-1437.) 
!      75-92918 

48-96403 
49-18559 
49-40715 
49-62870 

72- 

•25 
•5 

•75 

4071-50407 
409<J-82750 
4128-24909 
i    4156-76986 

2-26-19467 
226-i)8006 
227-76546 
228-55086 

<•>:)•>()>;;:) 
1    64-0-2989 
j     64-25145 
1    64-47300 

•25 

2J>5\)HS7 

76-71453 

49-85026 

73- 

418533681 

2-29-336-26 

i     64-69456 

•5 

2507-18728 

77-49998 

50-07182 

•25 

4214-10293 

230-1216* 

64-91612 

•75 
57- 

2520-42337 
2551-75363 

178-28538 
179-07078 

50-29337 
50-51493 

•5 

•75 

42  42-9  1722 
4-271  -88569 

230-90706 
231-69245 

1     65-13767 
65-35923 

•25 

•2574-19156 

179-85617 

50-73649 

74- 

4300-84034 

232-47785 

65-58079 

•5 

25'J6-7'2-267 

:     180-64157 

50-95804 

•25 

4329-94916 

233-26325 

6580234 

•75 

2619-35196 

1     181-42697 

51-17960 

•5 

4359-15615 

234-04865 

66-02390 

53- 

2642-07942 

18221237 

51-40116 

•75 

234-83405 

66-24546 

•25 

2664  90505 

182-99777 

51-62271 

75- 

4417-86466 

235-61944 

66-46701 

•5 

.  2687-82836 

183-78317 

5184427 

•25 

4447-36618 

236-40484 

66-63857 

•75 

2710-85084 

184-56855 

52-06583 

•5 

4476-96588 

237-19024 

,     66-91043 

59- 

2733-97100 

185-35396 

52-28738 

•75 

450666374 

237-97.''64 

67-13163 

•25 

2757-18933 

186-13936 

52-50894 

76- 

4536-45979 

238-76104 

67-35324 

•5 

2780-50584 

136-92476 

52-73050 

•25 

4566  35400 

239-54643 

67-57480 

•75 

280392053 

,     187-71016 

52-95205 

•5 

45^6  34640 

240-33183 

67-7-.'03.> 

60- 

28-27-43333 

,     1&3-49555 

53-17364 

•75 

46-26436116 

241-11723 

6301791 

•25 

2851-04442 

189-28095 

5339517 

77- 

465662571 

•241-9'!i63 

68-23J47 

•5 

2874-75362 

190-06635 

53-61*72 

•25 

463691262 

68-46102 

•75 

2-393-56100 

190-85175 

53-83828 

•5 

4717-29771 

243-47343 

6863258 

61- 

2922-46656 

191-63715 

54-05984 

•75 

4747-78093 

244-25.S8-2 

68-90414 

•25 

'2-J46-47029 

lii-2-42255 

54-28139 

78- 

477836242 

•245  044-22 

69-12570 

•5 

2970-57220 

193-20794 

54-50295 

•25 

4809-04204 

245-82962 

6931725 

•75 

2*4-772-23 

193-99334 

54-72451 

•5 

4839-81983 

246-6  ISM 

69-56831 

62- 

3019-07C54 

194-77874 

5494606 

•75 

4370-79579 

217-40042 

69-79037 

•25 

3iH34C6<J7 

195-56414 

55-16762: 

79- 

4901-66993 

248-18531 

70-01192 

•5 

3067-96157 

196-34954 

55-38918 

•25 

4i.i3-J-742-25 

248-97121 

70-23348 

•75 

309255J33 

197-13493 

55-61073 

•5 

4963-91274 

249-75661 

70-45504 

63- 

3117-24531 

197-92033 

55-83229 

•75 

4995-18140 

250-34201 

70-67659 

•25 

314-203144 

11-8-70573 

56-05385 

80- 

5026-54824 

251-32741 

70-89815 

3166-92174 

39949113 

56-27540 

•25 

5058-01325 

•252-11-j^l 

71-11971 

•75 

3191-907-22 

200-27653 

56-49696, 

•5 

5089-57644 

252-89820 

7  1-3}  126 

64- 

321G-99037 

201-06192 

56-71852 

•75 

512123781 

253-63360 

71-56282 

•25 

324-2-17270 

201-94732 

56-91007 

81- 

515-2-9ii735 

71-79438 

•5 

3267  -45-270 

202  63*72 

57-16163 

•25 

5184-85506 

S55-25440 

72-00593 

•~5 

3292-S3D33 

203-41312 

57-38319 

•5 

5216-81095 

2.Vj-<>3-J80 

72-2-2749 

65- 

3318-30724 

204-20352 

57-60475 

•75 

524H-86501, 

'2568-2579 

72-44905 

•25 

334388176 

204-98892 

57-82630: 

82- 

5-281-01725 

257-61059 

"2-67060 

•5 

•75 

3369-55447 
339532534 

205-77431 
206-55&71 

53-04786 
53-26942 

•25 
•5 

5313  26766 
5345616-24 

258-3^599 
'259-18139 

72-39216 
7311372 

66- 

3421-19439 

207-34511 

58-49097 

•75 

5378-06301 

259-%679 

7333527 

•25 

3447-1616-2 

208-13051 

53-71253 

83- 

5410-60794 

260-75219 

73-5563:? 

•5 

3473227u2 

208-91591 

58-93409 

•25 

5443-25105 

261-53758 

•75 

3499-39060 

209-70130 

59-15554 

•5 

5475-99234 

26-2-32298 

73-999V4 

67- 

3525-65235 

210-48670 

59-37720 

•75 

5508-83180 

263-10338 

74-22150 

•25 

3552-01228 

211-27210 

59-59876 

84- 

5541-76944 

•2o:<-s'.'.-J7> 

74-44306 

•5 

3578-47033 

212-05750 

59-82031 

•25 

5574-80525 

264-67918 

74-66461 

•75 

3605-02665 

212-84290 

60-04187 

•5 

5607-93923 

265-46457 

74-88617 

68- 

3631-68110 

213-62930 

60-26343 

•75 

5641-17139 

266-24997 

75-10773 

•25 

3658-43373 

214-41369 

60-43498' 

85- 

5674-50173 

267-03537 

75*3*^2ci 

•5 

3685-28453 

215-19909 

60-70654 

•25 

5707-93023 

267-82077 

75-55084 

•75 

3712-23350 

215-98449 

60-92810 

•5 

5741-45692 

868-606)7 

75-7724(1 

69- 

3-39-28065 

21676989 

61-14965 

•75 

5775-08178 

269-39157 

75-99395 

•25 
•5 

3766-42597 
3793-66947 

217-55529 
218-34068 

61-37121 

61-59277 

86- 
•25 

5308-80481; 
5342-6-2602 

270-17696 
270-96236 

76-21551 
76-43707 

•75 

33-21-01115 

219-12608 

61-81432 

•5 

5876-54340 

271-74776 

76-65302 

70 

3348-45100 

219-91148 

62-03538 

•75 

5910-56296 

27253316 

76-880  1« 

•25 

3875-98902 

220-69683 

62-25744 

87- 

5944  67369 

273-31356 

77-1017-I 

"5 

3903-62522 

S221-4.-I-2-2S 

62-47399 

•25 

5978-89260 

•274-103.'o 

77-:«3#j 

•75 

3931-35959 

222"2t>/63 

62-70055 

•5 

6013-20468 

274-88935: 

77-544H5 

71 

3959-19214 

223-05307 

62-92-2  11 

•75 

6047-6  141J4 

275-674751 

77-7664] 

•25 

3987-12286 

22383347 

03-1436(3 

88' 

6082-  1-2337 

276-46015 

;:•.'--.• 

32 


APPENDIX. 


|   Side  of 

Side  of 

Diara. 

Area. 

Circun).   j  equal  sq. 

Diam. 

Area. 

Circim. 

equal  tq. 

88-25 

6116-72993 

277-24555 

78-20952 

94-25 

6976-74097 

296-09510 

83-52688 

•5   6151-43476 

278-03094 

78-43103 

•5 

7012-80194 

296-88050 

83-74844 

•75  6186-23772 

278-81634 

78-65263 

•75 

7050-96109 

297-66590 

83-97000 

89-  1  6231-13885 

279-60174 

78-87419 

95- 

7083-21842 

298-45130 

84-19155 

•25 

6256-13815 

230-38714 

79-09575 

•25 

7125-57992 

299-23(370 

84-4131) 

•5 

6291-23563 

281-17251 

79-31730 

•5 

7163-02759 

300-02209 

84-63467 

•75 

6326-43129 

281-95794 

79-53886 

•75 

7200-57944 

300-807491  84-85622 

90- 

6361-72512 

282-74333 

79-76042 

96- 

7238-22947 

301-59239   85-07778 

•25 

6397-11712 

233-52873 

79-9819.3 

•25 

7275-97767 

302-37829   85-29934 

•5 

6432-60730 

281-31413 

80-20353 

•5 

7313-82404 

303-16369 

85-52089 

•75 

6469-19566 

285-09953 

80-42509 

•75 

7351-76359 

303-94908 

85-74245 

91- 

6503-8S219 

285-83493 

80-64669 

97- 

7339-81131 

304-73448 

85-96401 

.25 

653J-6G639 

286-67032 

80-86820 

•25 

7427-95221 

305-51983 

86-18556 

•5 

6575-51977 

237-45572 

81-08976 

•5 

7466-19129 

3!)6-30523 

86-40712 

•75 

6611-53J82 

288-24112 

81-31132 

•75 

7504-52853 

307-09068 

86-62868 

92- 

6547-61005 

239-02652 

81-53287 

98- 

7542-96396 

307-87603 

86-85023 

•25 

6633-78745 

289-31192 

81-75413 

•25 

7581-49755 

308-66147 

87  -07175; 

•5 

672;)-06303 

aso-sarsa 

81-975.>'J 

•5 

7620-12933 

309-44637 

87-29335 

•75 

6756-43678 

•291-34271 

8-2-1  '.'754 

•75 

7653-H5927 

310-23227 

87-51490 

93- 

6792-90871 

292-16811 

82-41910 

99- 

7697-68739 

311-01767 

87-73646 

•25 

6829-47831 

292-95351 

82  -040(56 

•25 

7736-61369 

311-80307 

87-95802 

•5 

6866-14709 

293-73391 

82-86221 

•5 

7775-63816 

312-58846 

88-17957 

•75 
94- 

6903-91354  2H4-5243I 
6939-77817  29530970 

83-i  W377 
H:>  30533! 

•75 
100- 

7814-76031 
7353-98163 

313-373361  88-40113 
314-15926!  88-62269 

The  following  rules  are  for  extending  the  use  of  the  above  table. 

To  fnd  the  area,  circumference,  or  side  of  equal  square,  of  a  circle 
having  a  diameter  of  more  than  100  inches,  feet,  tyc.  Rule. — Divide 
the  given  diameter  by  a  number  that  will  give  a  quotient  equal  to  some 
one  of  the  diameters  in  the  table  ;  then  the  circumference  or  side  of 
equal  square,  opposite  that  diameter,  multiplied  by  that  divisor,  or,  the 
area  opposite  that  diameter,  multiplied  by  the  square  of  the  aforesaid 
divisor,  will  give  the  answer. 

Example* — What  is  the  circumference  of  a  circle  whose  diameter  is 
228  feet  ?    228,  divided  by  3,  gives  76,  a  diameter  of  the  table,  the  cir- 
cumferejice  of  which  is  238-761,  therefore  :  . 
238-761 
3 


716-283  feet.  Ans. 

Another  example. — What  is  the  area  of  a  circle  having  a  diameter 
of  150  inches  ?    150,  divided  by  10,  gives  15,  one  of  the  diameters  in 
the  table,  the  area  of  which  is  176-71458,  therefore  : 
176-71458 

100  —  10  X  10 


17,671-45800  inches.  Ans. 

To  find  the  area,  circumference,  or  side  of  equal  square,  of  a  circle 
Having  an  intermediate  diameter  to  those  in  the  table.  Rule. — Multiply 
the  given  diameter  by  a  number  that  will  give  a  product  equal  to  some 
one  of  the  diameters  in  the  table  ;  then  the  circumference  or  side  of 
equal  square  opposite  that  diameter,  divided  by  that  multiplier,  or,  the 
area  opposite  that  diameter  divided  by  the  equate  of  the  aforesaid  mul 
tiplier,  will  give  the  answer. 


APPENDIX. 

Example. — What  is  the  circumference  of  a  circle  whose  diameter  is 
6$,  or  6-125  inches  ?     6-125,  multiplied  by  2,  gives  12-25,  one  of  the 
diameters  of  the  tablo,  whose  circumference  is  38-484,  therefore  : 
2)38-484 

19-242  inches.  Ans. 

Another  example. — What  is  the  area  of  a  circle,  the  diameter  of 
which  is  3-2  feet  ?  3-2,  multiplied  by  5,  gives  16,  and  the  area  of  16 
is  201-0619,  therefore: 

5  X  5  —  25)201-0619(8-0424  4-  feet.  Ans. 
200 

106 
100 

61 
50 

119 
100 

19 

Note. — The  diameter  of  a  circle,  multiplied  by  3-14159,  will  give 
Us  circumference  ;  the  square  of  the  diameter,  multiplied  by  -78539, 
will  give  its  area  ;  and  the  diameter,  multiplied  by  -88622,  will  give 
the  side  of  a  square  equal  to  the  area  of  the  circle. 


TABLE    SHOWING   THE    CAPACITY  OF  WELLS,  CISTBRNS,  AC. 

The  gallon  of  the  State  of  New  York,  by  an  act  passed  April  11, 1851,  Is  required  to  conform 
to  the  standard  gallon  of  the  United  States  government.  This  standard  gallon  contains  231«nbi« 
inches.  In  conformity  with  this  standard  the  following  table  has  been  computed. 

One  foot  in  depth  of  a  cistern  of 

3  feet  diameter  will  contain        ....        62-872  gallon* 
31.         «  « 71-965      " 

4  «  « 93-995  " 

4A  "  " 118-963  « 

5  «  « 146-868  " 

54-  «  " 177-710  " 

6  «  « 211-490  " 

61  «  « 248-207  " 

7  «  « 287-861  " 

8  «  « 375-982  " 

9  "                       ".....       475-852       " 
10            «                       «                   ....       587-472       " 
12  «  « 845-959       '• 

y0te  —To  reduce  cubic  feet  to  gall  >ns,  multiply  by  7-48. 


TAB!  E  OF  POLYGONS. 

'From  Gregory's  Mathematics.) 


!-l 
*H 

Names. 

Multipliers  for 
areas. 

Kadim  of  cir- 
cum.  circle. 

Factors  for 
sides. 

3 

Trigon 

0-4830127 

0-5773503 

1-732051 

4 

Tetragon,  or  Square 

!•  0.000000 

0-7071068 

1-414214 

5 

Pentagon    - 

1-7204774 

0-8506508 

1-175570 

6 

Hexagon 

2-5980762 

1-0000000 

1-000000 

:   7 

Heptagon  - 

3-6339124 

1-1523824 

0-867767 

8 

^Octagon 

4-8284271 

1-3065628 

0-765367 

9' 

Nonagon    - 

6-1818242 

1-4619022 

0-684040 

10 

Decagon 

7-6942088 

1-6180340 

0-618034 

11 

Undecagon 

9-3656399 

1-7747324 

0-563465 

12 

Dodecagon 

11-1961524 

1-9318517 

0-517638 

To  find  the  area  of  any  regular  polygon,  whose  sides  do  not  exceed 
twelve.  Rule. — Multiply  the  square  of  a  side  of  the  given  polygon  by 
the  number  in  the  column  termed  Multipliers  for  areas,  standing  op 
posite  the  name  of  the  given  polygon,  and  the  product  will  be  the  an- 
swer. Example. — What  is  the  area  of  a  regular  heptagon,  whose 
sides  measure  each  2  feet  ? 

3-6339124 

4  =  2X2 


14-5356496:  Ans. 

To  find  the  radius  of  a  circle  which  will  circumscribe  any  regular 
polygon  given,  whose  sides  do  not  exceed  twelve.  Rule. — Multiply  a 
side  of  the  given  polygon  by  the  number  in  the  column  termed  Radius 
of  circumscribing  circle,  standing  opposite  the  name  of  the  given  poly- 
gon,  and  the  product  will  give  the  answer.  Example. — What  is  the 
radius  of  a  circle  which  will  circumscribe  a  regular  pentagon,  whose 
sides  measure  each  10  feet  ? 

•8506508 
10 


8-5065080:  Ans. 

To  find  the  side  of  any  regular  polygon  that  may  be  inscribed  within 
a  given  circle.     Rule. — Multiply  the  radius  of  the  given  circle  by  the 
number  in  the  column  termed  Factors  for  sides,  standing  opposite  the 
name  of  the  given  polygon,  and  the  product  will  be  the  answer.     Ex- 
ample.— What  is  the  side  of  a  regular  octagon  that  may  be  inscribed 
within  a  circle,  whose  radius  is  5  feet  ? 
•765367 
5 


3-826335:  Ans. 


WEIGHT  OF  MATERIALS 


,.,     ,                                             His.  m  o 
Woods.                                        cubic  foot. 

Metals.                              eJ&jto. 

Apple,      -         -        -         -     49 

Wire-drawn  brass,           -     534 

Ash,                                          4D 

Cast  brass,  '        -         -         506 

Beach,     -         .        .         .40 

Sheet-copper,          .        .     549 

Birch,            ...        45 

Pure  cast  gold,    •                1210 

Box,         ....     60 

Bar-iron,          -           475  to  487 

Cedar,                                      28 

Cast  iron,    .         -      450  to  475 

Virginian  red  cedar,          -     40 

Milled  lead,    -         -         -     713 

Cherry,          ...         38 

Cast  lead,             -         -         709 

Sweet  chestnut,         -         -     36 

Pewter,           ...     453 

Horse-chestnut,                        34 

Pure  platina,       -         -       1345 

Cork,        -         -         .         .15 

Pure  cast  silver,      -         -     654 

,  I  'ypress,                                     28 
Ebony,     -         -         -         -83 

Steel,           -         -       486  to  490 
Tin,        -         -         -         -     456 

Elder,            ...         43 

Zinc,                                        439 

Elm,         -         .         .         -     34 

Stone,  Earths,  fyc. 

Fir,  (white  spruce,)        -         29 

Brick,  Phila.  stretchers,       105 

Hickory,           -         -         -     52 

North  river  common  hard 

Lance-wood,          -         -         59 

brick,           -         -         -     107 

Larch,     -         -         -         -     31 

Do.             salmon  brick,     100 

Larch,  (whitewood,)       -         22 

Brickwork,  about         -           95 

Lignum-vitae,    -         -         -     83 

Cast  Roman  cement,        -     100 

Logwood,      -         -         -         57 

Do.  and  sand  in  equal  parts,  113 

St.  Domingo  mahogany,     -     45 

Chalk,         -                 144  to  166 

Honduras,  or  bay  mahogany,  35 
Maple,                                      47 

Clay,      -         -         -         -     11& 
Potter's  clay,       -       112  to  130 

White  oak,       -         -     43  to  53 

Common  earth,             95  to  124 

Canadian  oak,                          54 

Flint,      .-.         -     163 

Red  oak,                              -     47 

Plate-glass,          •         -         172 

Live  oak,                                 76 

Crown-glass,  -                        157 

White  pine,      -         -     23  to  30 

Granite,      -         -       158  to  187 

Yellow  pine,                   34  to  44 

Quincy  granite,       -         -     166 

Pitch  pine,        -         -    46  to  58 

Gravel,                                    109 

Poplar,                                       25 

Grindstone,     -         -         -     134 

Sycamore,        -         -        -     36 

Gvpsum,  (Plaster-stone,)       142 

Walnut,        -        -                  40 

Unslaked  lime,    -         -           53 

36 


APPENDIX. 


Hi*,  in  a 
culiic  foot. 

Limestone,      -         -  118  to  198 

Common  blue  stone, 

Marble,       -         -       161  to  177 

Silver-gray  flagging, 

New  mortar,  -         -         -     107 

Stonework,  about, 

Dry  mortar,          -         -           90 

Common  plain  tiles, 

Mortar  with  hair,  (Plaster- 

Sundries, 

ing,)  ....     105 

Atmospheric  air, 

Do.     dry,           -         •           86 

Yellow  beeswax,     - 

Do.     do.     including  lath 

Birch-charcoal,    - 

and  nails,  from  7  to   11 

Oak-charcoal, 

Ibs.  per  superficial  foot. 
Crystallized  quartz,          -     165 

Pine-charcoal,      - 
Solid  gunpowder,     - 

Pure  quartz-sand,         -         171 

Shaken  gunpowder, 

Clean  and  coarse  sand,          100 

Honey, 

Welsh  slate,   -         -         -     180 

Milk, 

Paving  stone,       -         -         151 

Pitch,     - 

Pumice  stone,          -                56 

Sea-water, 

Nyack  brown  stone,     -         148 

Rain-water,    - 

Connecticut  brown  stone,       170 

Snow, 

Tarrytown  blue  stone,     -     171 

Wood-ashes,  * 

Ibs.  in  a. 


160 
185 
120 
115 

0-075 
60 
34 
21 
17 

.  109 
58 

•  90 
64 

.  71 
64 

.  62-5 

8 

53 


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98 


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